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Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 289854 9 pageshttpdxdoiorg1011552013289854
Research ArticleThe Monoid Consisting of Kuratowski Operations
Szymon Plewik and Marta WalczyNska
Institute of Mathematics University of Silesia ul Bankowa 14 40-007 Katowice Poland
Correspondence should be addressed to Szymon Plewik plewikmathusedupl
Received 30 August 2012 Accepted 1 November 2012
Academic Editor Nan-Jing Huang
Copyright copy 2013 S Plewik and M WalczynskaThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
The paper fills gaps in knowledge about Kuratowski operations which are already in the literature The Cayley table for theseoperations has been drawn up Techniques using only paper and pencil to point out all semigroups and its isomorphism typesare applied Some results apply only to topology and one cannot bring them out using only properties of the complement and aclosure-like operation The arguments are by systematic study of possibilities
1 Introduction
Let119883 be a topological space Denote by 119860minus closure of the set119860 sube 119883 Let 119860119888 be the complement of 119860 that is 119883 119860 =
119860119888 The aim of this paper is to examine monoids generated
under compositions from the closure and the complement Awidely known fact due to Kuratowski [1] states that at most 14distinct operations can be formed from such compositionsMark them as follows Kuratowski operations
1205900
(119860) = 119860 (the identity)1205901
(119860) = 119860119888 (the complement)
1205902
(119860) = 119860minus (the closure)
1205903
(119860) = 119860119888minus
1205904
(119860) = 119860minus119888
1205905
(119860) = 119860119888minus119888 (the interior)
1205906
(119860) = 119860minus119888minus
1205907
(119860) = 119860119888minus119888minus
1205908
(119860) = 119860minus119888minus119888
1205909
(119860) = 119860119888minus119888minus119888
12059010
(119860) = 119860minus119888minus119888minus
12059011
(119860) = 119860119888minus119888minus119888minus
12059012
(119860) = 119860minus119888minus119888minus119888
12059013
(119860) = 119860119888minus119888minus119888minus119888
The following rules was found in the original paper byKuratowski [1 pages 183-184] In the Engelking book [2] theyare commented by a hint on page 81 Cancellation rules
119860minus119888minus
= 119860minus119888minus119888minus119888minus
119860119888minus119888minus
= 119860119888minus119888minus119888minus119888minus
(1)
Kuratowski operations have been studied by severalauthors for example [3] or [4] A list of some other authorsone can find in the paper [5] by Gardner and Jackson For thefirst time these operations were systematically studied in thedissertation by Kuratowski whose results were published in[1] Tasks relating to these operations are usually resolved atlectures or exercises with general topologyThey are normallyleft to students for independent resolution For exampledetermine how many different ways they convert a given set
This note is organized as follows Kuratowski operationsand their marking are described in the introduction Theirproperties of a much broader context than for topologies arepresented in Section 2The Cayley table for the monoidM ofall Kuratowski operations has been drawn up in Section 3We hope that this table has not yet been published in theliterature Having this table one can create a computerprogram that calculates all the semigroups contained in MHowever in Sections 4ndash8 we present a framework (ietechniques using only paper and pencil) to point out all 118semigroups and 56 isomorphism types of themThe list of 43semigroups which are not monoids is presented in Section 9In this section also isomorphism types are discussed in orderof the number of elements in semigroups Finally we present
2 Journal of Mathematics
cancellation rules (relations) motivated by some topologicalspaces
2 Cancellation Rules
A map 119891 119875(119883) rarr 119875(119883) is called
(i) increasing if 119860 sube 119861 implies 119891(119860) sube 119891(119861)(ii) decreasing if 119860 sube 119861 implies 119891(119861) sube 119891(119860)(iii) an involution if the composition 119891 ∘ 119891 is the identity(iv) an idempotent if 119891 ∘ 119891 = 119891
Assume that 119860 997891rarr 1205900
(119860) is the identity 119860 997891rarr 1205901
(119860) isa decreasing involution and 119860 997891rarr 120590
2
(119860) is an increasingidempotent map Other operations 120590
119894
let be compositions of1205901
and 1205902
as it was with the kuratowski operationsWe get thefollowing cancellation rules
Lemma 1 If 119861 sube 1205902
(119861) then
1205902
∘ 12059012
= 1205906
1205902
∘ 12059013
= 1205907
(2)
Proof For clarity of this proof use designations 1205901
(119860) = 119860119888
and 1205902
(119860) = 119860minus Thus we shall prove
119860minus119888minus119888minus119888minus
= 119860minus119888minus
119860119888minus119888minus119888minus119888minus
= 119860119888minus119888minus
(3)
We start with 119860minus119888minus119888 sube 119860minus119888minus119888minus substituting 119861 = 119860minus119888minus119888 in 119861 sube119861minus This corresponds to 119860minus119888minus119888minus119888 sube 119860minus119888minus119888119888 = 119860minus119888minus since 120590
1
isa decreasing involution Hence 119860minus119888minus119888minus119888minus sube 119860minus119888minus since 120590
2
anincreasing idempotent
Since 1205901
is decreasing 1205902
is increasing and 119861 sube 119861minus wehave
119861minus119888
sube 119861119888
sube 119861119888minus
(4)
Thus119860minus119888minus119888 sube 119860minus119888119888minus = 119860minus if we put119860minus119888 = 119861 Again using that1205902
is increasing and 1205901
is decreasing we obtain 119860minus119888minus119888minus sube 119860minusand then 119860minus119888 sube 119860minus119888minus119888minus119888 Finally we get 119860minus119888minus sube 119860minus119888minus119888minus119888minus
With 119860119888 in the place of 119860 in the rule 119860minus119888minus = 119860minus119888minus119888minus119888minus weget the second rule
In academic textbooks of general topology for example[2 Problem 171] one can find a hint suggested to prove theabove cancellation rules Students go like this steps 119860 sube 119860
minus
and 119860119888 sube 119860119888minus lead to 119860119888minus119888 sube 119860 the special case 119860minus119888minus119888minus119888 sube
119860minus119888minus (of119860119888minus119888 sube 119860) leads to119860minus119888minus119888minus119888minus sube 119860minus119888minus steps119860minus119888minus119888 sube 119860minus
and 119860minus119888minus119888minus sube 119860minus lead to 119860minus119888minus sube 119860minus119888minus119888minus119888minus at the end use thelast step of the proof of Lemma 1 Note that the above proofsdo not use the axioms of topology as follows
(i) 0 = 0minus(ii) (119860 cup 119861)minus = 119860minus cup 119861minus
In the literature there are articles in which Kuratowski oper-ations are replaced by some other mappings For exampleKoenen [6] considered linear spaces and put 120590
2
(119860) to bethe convex hull of 119860 In fact Shum [7] considered 120590
2
as theclosure due to the algebraic operations Add to this that theseoperations can be applied to so-called Frechet (V) spaceswhich were considered in the book [8 pages 3ndash37]
3 The Monoid M
LetM be the monoid consisting of all Kuratowski operationsthat is there are assumed cancellation rules 120590
6
= 1205902
∘12059012
and1205907
= 1205902
∘ 12059013
Fill in the Cayley table for M where the rowand column marked by the identity are omitted Similarly asin [9] the factor that labels the row comes first and that thefactor that labels the column is second For example 120590
119894
∘ 120590119896
isin the row marked by 120590
119894
and the column marked by 120590119896
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
12059010
12059011
12059012
12059013
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
12059012
12059013
12059010
12059011
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205903
1205902
1205906
1205907
1205902
1205903
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
1205905
1205904
1205908
1205909
1205904
1205905
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
1205907
1205906
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
1205909
1205908
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
12059010
12059011
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
12059011
12059010
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
12059012
12059013
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
12059013
12059012
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
(5)
It turns out that the above table allows us to describeall semigroups contained inM using pencil-and-paper tech-niques only The argument will be by a systematic study ofpossibilities Preparing the list of all semigroups consisting ofKuratowski operations we used following principles
(i) minimal collection of generators is written using⟨119860 119861 119885⟩ where letters denote generators
(ii) when a semigroup has a few minimal collections ofgenerators then its name is the first collection in thedictionary order
(iii) all minimal collections of generators are written withthe exception of some containing 120590
0
(iv) we leave to the readers verification that our list is
complete sometimes we add hints
4 Semigroups with 1205901
Observe that each semigroup which contains 1205900
is a monoidSince120590
1
∘ 1205901
= 1205900
a semigroupwhich contains1205901
is amonoidtoo
Theorem 2 There are three monoids containing 1205901
(1) ⟨1205901
⟩ = 1205900
1205901
(2) M = ⟨120590
1
120590119894
⟩ = 1205900
1205901
12059013
where 119894 isin 2 3 4 5(3) letM
1
= ⟨1205901
1205906
⟩ If 119895 isin 6 7 13 then
M1
= ⟨1205901
120590119895
⟩ = 1205900
1205901
cup 1205906
1205907
12059013
(6)
Proof The equality ⟨1205901
⟩ = 1205900
1205901
is obviousSince 120590
2
= 1205903
∘ 1205901
= 1205901
∘ 1205904
= 1205901
∘ 1205905
∘ 1205901
we have⟨1205901
1205902
⟩ = ⟨1205901
1205903
⟩ = ⟨1205901
1205904
⟩ = ⟨1205901
1205905
⟩ = MIf 119895 isin 6 7 13 then any composition 120590
119895
∘ 120590119894
or 120590119896
∘ 120590119895
belongs to M1
and so ⟨1205901
120590119895
⟩ sube M1
We have 1205907
= 1205906
∘ 1205901
Journal of Mathematics 3
1205908
= 1205901
∘ 1205906
1205909
= 1205901
∘ 1205907
12059010
= 1205906
∘ 1205906
12059011
= 1205906
∘ 1205907
12059012
= 1205908
∘ 1205906
and 12059013
= 1205908
∘ 1205907
and so M1
= ⟨1205901
1205906
⟩ =
1205900
1205901
cup 1205906
1205907
12059013
Since 120590
6
= 1205907
∘ 1205901
= 1205901
∘ 1205908
= 1205901
∘ 1205909
∘ 1205901
= 12059010
∘ 1205901
∘ 12059010
=
12059011
∘ 12059011
∘ 1205901
= 1205901
∘ 12059012
∘ 12059012
and 1205906
= 1205901
∘ 12059013
∘ 1205901
∘ 12059013
∘ 1205901
we have ⟨120590
1
120590119895
⟩ = M1
for 119895 isin 6 7 13
Consider the permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
12059010
12059011
12059012
12059013
1205900
1205901
1205905
1205904
1205903
1205902
1205909
1205908
1205907
1205906
12059013
12059012
12059011
12059010
)
(7)
It determines an automorphism A M rarr M
Theorem 3 The identity and A are the only automorphismsofM
Proof Delete rows and columns marked by 1205901
in the Cayleytable for M Then check that the operation 120590
3
is in the rowor the column marked by 120590
3
only Also the operation 1205904
isin the row or the column marked by 120590
4
only Therefore thesemigroup
⟨1205903
1205904
⟩ = 1205902
1205903
12059013
(8)
has a unique minimal set of generators 1205903
1205904
The reader isleft to check this with the Cayley table forM
Suppose G is an automorphism of M By Theorem 2 Gtransforms the set 120590
2
1205903
1205904
1205905
onto itself However 1205902
and1205905
are idempotents but 1205903
and 1205904
are not idempotents Sothere are two possibilities G(120590
3
) = 1205903
and G(1205904
) = 1205904
whichimplies that G is the identity G(120590
3
) = 1205904
and G(1205904
) = 1205903
which implies G = A The reader is left to check this with theCayley table forM We offer hints 120590
2
= 1205903
∘ 1205904
1205905
= 1205904
∘ 1205903
1205906
= 1205903
∘ 1205902
1205907
= 1205903
∘ 1205903
1205908
= 1205904
∘ 1205904
1205909
= 1205904
∘ 1205905
12059010
= 1205906
∘ 1205906
12059011
= 1205906
∘ 1205907
12059012
= 1205908
∘ 1205906
and 12059013
= 1205908
∘ 1205907
to verify the details of this proof
5 The Monoid of All Idempotents
The set 1205900
1205902
1205905
1205907
1205908
12059010
12059013
consists of all squares inMThese squares are idempotents and lie on the main diagonalin the Cayley table forM They constitute a monoid and
1205900
1205902
1205905
1205907
1205908
12059010
12059013
= ⟨1205900
1205902
1205905
⟩ (9)
The permutation
(1205900
1205902
1205905
1205907
1205908
12059010
12059013
1205900
1205902
1205905
1205908
1205907
12059010
12059013
) (10)
determines the bijection I ⟨1205900
1205902
1205905
⟩ rarr ⟨1205900
1205902
1205905
⟩ suchthat
I (120572 ∘ 120573) = I (120573) ∘ I (120572) (11)
for any 120572 120573 isin ⟨1205900
1205902
1205905
⟩ To verify this apply equalities1205902
∘ 1205905
= 1205907
1205905
∘ 1205902
= 1205908
1205902
∘ 1205905
∘ 1205902
= 12059010
and1205905
∘ 1205902
∘ 1205905
= 12059013
Any bijection I 119866 rarr 119867 having propertyI(120572∘120573) = I(120573)∘I(120572) transposesCayley tables for semigroups119866
and119867 Several semigroups contained inM have this propertyWe leave the reader to verify this
We shall classify all semigroups contained in the semi-group ⟨120590
2
1205905
⟩ Every such semigroup can be extended toa monoid by attaching 120590
0
to it This gives a completeclassification of all semigroups in ⟨120590
0
1205902
1205905
⟩The semigroup ⟨120590
2
1205905
⟩ contains six groups with exactlyone element
Semigroups ⟨1205902
12059010
⟩ = 1205902
12059010
and ⟨1205905
12059013
⟩ = 1205905
12059013
aremonoids Both consist of exactly two elements and are notgroups so they are isomorphic
Semigroups ⟨1205907
12059010
⟩ and ⟨1205908
12059013
⟩ are isomorphic in par-ticular A[⟨120590
7
12059010
⟩] = ⟨1205908
12059013
⟩ Also semigroups ⟨1205907
12059013
⟩
and ⟨1205908
12059010
⟩ are isomorphic by A Every of these four semi-groups has exactly two elements None of them is a monoidThey form two types of nonisomorphic semigroups becausethe bijections (120590
7
12059010
) (12059010
1205908
) and (1205907
1205908
) (12059010
12059010
) arenot isomorphisms
Semigroups ⟨1205902
1205907
⟩ = 1205902
1205907
12059010
and ⟨1205905
1205908
⟩ =
1205905
1205908
12059013
are isomorphic Also semigroups ⟨1205902
1205908
⟩ =
1205902
1205908
12059010
and ⟨1205905
1205907
⟩ = 1205905
1205907
12059013
are isomorphic Infact A[⟨120590
2
1205907
⟩] = ⟨1205905
1205908
⟩ and A[⟨1205902
1205908
⟩] = ⟨1205905
1205907
⟩None of these semigroups is a monoid They form two typesof nonisomorphic semigroups Indeed any isomorphismbetween ⟨120590
2
1205907
⟩ and ⟨1205902
1205908
⟩ must be the identity on themonoid ⟨120590
2
12059010
⟩ Therefore would have to be the restrictionof I But I restricted to ⟨120590
7
12059010
⟩ is not an isomorphismSemigroups ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
and ⟨1205905
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
are notmonoids They are isomorphic by A
The semigroup ⟨1205902
1205905
⟩ contains exactly one semigroupwith four elements ⟨120590
7
1205908
⟩ = ⟨12059010
12059013
⟩ = 1205907
1205908
12059010
12059013
which is not a monoidNote that ⟨120590
2
1205905
⟩ contains twenty different semigroupswith nine non-isomorphism types These are six isomorphicgroups with exactly one element ⟨120590
2
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong
⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ two isomorphic monoids with exactlytwo elements ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ two pairs of isomorphicsemigroups with exactly two elements ⟨120590
7
12059010
⟩ cong ⟨1205908
12059013
⟩
and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ two pairs of isomorphic semigroupswith exactly three elements ⟨120590
2
1205907
⟩ cong ⟨1205905
1205908
⟩ and ⟨1205902
1205908
⟩ cong
⟨1205905
1205907
⟩ a semigroup with exactly four elements ⟨1205907
1205908
⟩and two isomorphic semigroups with exactly five elements⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ and also ⟨1205902
1205905
⟩ Thus ⟨1205902
1205905
⟩ containstwenty different semigroups with nine isomorphism typesBut ⟨120590
0
1205902
1205905
⟩ contains forty-one different semigroups withseventeen isomorphism types Indeed adding 120590
0
to semi-groups contained in ⟨120590
2
1205905
⟩ which are not monoids weget twenty monoids with eight non-isomorphism typesAdding 120590
0
to a group contained in ⟨1205902
1205905
⟩ we get a monoidisomorphic to ⟨120590
2
12059010
⟩
6 The Semigroup Consisting of 12059061205907 120590
13
Using the Cayley table forM check thatA [1205906
1205907
12059013
] = 1205906
1205907
12059013
(12)Similarly check that the semigroup ⟨120590
6
1205909
⟩ = 1205906
1205907
12059013
can be represented as ⟨1205906
12059013
⟩ ⟨1205907
12059012
⟩ ⟨1205908
12059011
⟩
4 Journal of Mathematics
⟨1205909
12059010
⟩ or ⟨12059011
12059012
⟩ Also ⟨1205906
1205909
⟩ = ⟨1205906
1205907
1205908
⟩ =
⟨1205907
1205908
1205909
⟩ = ⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590
6
1205909
⟩ Other semigroupsincluded in ⟨120590
6
1205909
⟩ have one or two minimal collectionof generators One generator has groups ⟨120590
6
⟩ = 1205906
12059010
⟨1205909
⟩ = 1205909
12059013
⟨12059011
⟩ = 1205907
12059011
and ⟨12059012
⟩ = 1205908
12059012
Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590
7
12059010
⟩ ⟨1205907
12059013
⟩ ⟨1205908
12059010
⟩ ⟨1205908
12059013
⟩ and⟨1205907
1205908
⟩ are discussed in the previous section Contained in⟨1205906
1205909
⟩ and not previously discussed semigroups are ⟨1205906
1205907
⟩⟨1205906
1205908
⟩⟨1205907
1205909
⟩ and ⟨1205908
1205909
⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590
6
1205909
⟩ but different from thewhole One has
(i) ⟨1205906
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(ii) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(iii) ⟨120590
7
1205909
⟩ = ⟨1205909
12059011
⟩ = ⟨12059011
12059013
⟩ = 1205907
1205909
12059011
12059013
(iv) ⟨120590
8
1205909
⟩ = ⟨1205909
12059012
⟩ = ⟨12059012
12059013
⟩ = 1205908
1205909
12059012
12059013
Proposition 4 Semigroups ⟨1205906
1205907
⟩ and ⟨1205908
1205909
⟩ are isomor-phic and also semigroups ⟨120590
6
1205908
⟩ and ⟨1205907
1205909
⟩ are isomorphicbut semigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ are not isomorphic
Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906
1205907
⟩ rarr ⟨1205906
1205908
⟩ is an isomorphism Thus 119869[⟨1205907
12059010
⟩] =
⟨1205908
12059010
⟩ Given 119869(1205907
) = 1205908
we get 12059010
= 119869(12059010
) = 119869(1205907
∘
12059010
) = 1205908
∘ 12059010
= 1205908
But 119869(1205907
) = 12059010
implies 12059010
= 119869(1205907
) =
119869(12059010
∘ 1205907
) = 1205908
∘ 12059010
= 1205908
Both possibilities lead to acontradiction
So ⟨1205906
1205909
⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590
7
⟩ cong ⟨1205908
⟩ cong
⟨12059010
⟩ cong ⟨12059013
⟩ four isomorphic groups with exactly twoelements ⟨120590
6
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ and five semigroupswith exactly four elements ⟨120590
6
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong
⟨1205907
1205909
⟩ ⟨1205907
1205908
⟩ and also ⟨1205906
1205909
⟩
7 Remaining Semigroups in ⟨1205903 1205904⟩
We have yet to discuss semigroups included in ⟨1205903
1205904
⟩ notincluded in ⟨120590
2
1205905
⟩ and containing at least one of Kuratowskioperation 120590
2
1205903
1205904
or 1205905
It will be discussed up to theisomorphism A Obviously ⟨120590
2
⟩ = 1205902
and ⟨1205905
⟩ = 1205905
aregroups
71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590
2
1205906
⟩ = 1205902
1205906
12059010
and ⟨1205902
12059010
⟩ = 1205902
12059010
havedifferent numbers of elements Also ⟨120590
2
1205906
⟩ is isomorphic to⟨1205900
1205906
⟩ Nonisomorphic semigroups ⟨1205902
1205907
⟩ and ⟨1205902
1205908
⟩ arediscussed above The following three semigroups
(i) ⟨1205902
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(ii) ⟨1205902
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(iii) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ or ⟨1205907
1205908
⟩ This isimpossible by Proposition 4 and because ⟨120590
7
1205908
⟩ consists ofidempotents but 120590
6
is not an idempotent The nine-elementsemigroup on the set 120590
2
1205906
1205907
12059013
is represented as⟨1205902
1205909
⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590
2
12059011
⟩ ⟨1205902
12059012
⟩ ⟨1205902
12059013
⟩ and ⟨1205902
1205909
⟩Using A we have described eight semigroupsmdasheach
one isomorphic to a semigroup previously discussedmdashwhichcontains 120590
5
and elements (at least one) of ⟨1205906
1205909
⟩ Collec-tions of generators ⟨120590
2
1205906
12059013
⟩ ⟨1205902
1205907
12059012
⟩ ⟨1205902
1205908
12059011
⟩⟨1205902
12059011
12059012
⟩ ⟨1205902
12059011
12059013
⟩ and ⟨1205902
12059012
12059013
⟩ are minimalin ⟨120590
2
1205909
⟩ Also collections of generators ⟨1205905
1205907
12059012
⟩⟨1205905
1205908
12059011
⟩ ⟨1205905
1205909
12059010
⟩ ⟨1205905
12059010
12059011
⟩ ⟨1205905
12059010
12059012
⟩ and⟨1205905
12059011
12059012
⟩ are minimal in ⟨1205905
1205906
⟩
72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590
3
⟩ = 1205903
1205907
12059011
and ⟨1205904
⟩ =
1205904
1205908
12059012
are isomorphic by A They are not monoids Thesemigroup ⟨120590
3
⟩ can be extended using elements of ⟨1205906
1205909
⟩in three following ways
(i) ⟨1205903
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(ii) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(iii) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
Semigroups ⟨1205903
1205906
⟩ and ⟨1205903
1205909
⟩ are not isomorphic Indeedsuppose 119869 ⟨120590
3
1205906
⟩ rarr ⟨1205903
1205909
⟩ is an isomorphismThus 119869 isthe identity on ⟨120590
3
⟩ 119869(1205906
) = 1205909
and 119869(12059010
) = 12059013
This givesa contradiction since 120590
3
∘ 1205906
= 12059010
and 1205903
∘ 1205909
= 1205907
Neither ⟨120590
3
1205906
⟩ nor ⟨1205903
1205909
⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590
2
1205909
⟩ is not isomorphic to ⟨1205903
1205908
⟩since ⟨120590
2
1205909
⟩ has a unique pair of generators and ⟨1205903
1205908
⟩ =
⟨1205903
12059012
⟩Using A we getmdashisomorphic to previously discussed
onesmdashsemigroups ⟨1205904
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
⟨1205904
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
and ⟨1205904
1205907
⟩ =
⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
There exist minimal collec-tions of generators such as follows
⟨1205903
1205908
⟩ = ⟨1205903
1205906
1205909
⟩ = ⟨1205903
1205906
12059013
⟩
= ⟨1205903
1205909
12059010
⟩ = ⟨1205903
12059010
12059013
⟩
⟨1205904
1205907
⟩ = ⟨1205904
1205906
1205909
⟩ = ⟨1205904
1205909
12059010
⟩
= ⟨1205904
1205906
12059013
⟩ = ⟨1205904
12059010
12059013
⟩
(13)
73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590
2
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
and⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
= A[⟨1205902
1205903
⟩] andalso ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
and ⟨1205903
1205905
⟩ =
1205903
1205905
1205907
1205909
12059011
12059013
= A[⟨1205902
1205904
⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types
Journal of Mathematics 5
Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590
2
1205905
⟩ has 6 elements which are idempotentsIn ⟨1205903
1205904
⟩ there are seven semigroups which have threegenerators and have not two generators These are
(1) ⟨1205902
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩
(2) ⟨1205904
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩
(3) ⟨1205902
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩
(4) ⟨1205903
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩
(5) ⟨1205902
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩
(6) ⟨1205902
1205903
1205905
⟩
(7) ⟨1205902
1205904
1205905
⟩
8 Semigroups which Are Contained in ⟨12059031205904⟩
Groups ⟨1205902
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong ⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ have oneelement and are isomorphic
Groups ⟨1205906
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ monoids ⟨1205902
12059010
⟩ cong
⟨1205905
12059013
⟩ and also semigroups ⟨1205907
12059010
⟩ cong ⟨1205908
12059013
⟩ and⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ have two elements and Cayley tables asfollows
119860 119861
119860 119861 119860
119861 119860 119861
119860 119861
119860 119860 119861
119861 119861 119860
119860 119861
119860 119860 119861
119861 119860 119861
119860 119861
119860 119860 119860
119861 119861 119861
(14)
Monoids ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩ and also semigroups⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong ⟨1205904
⟩ havethree elements and Cayley tables as follows
119860 119861 119862
119860 119860 119861 119862
119861 119861 119862 119861
119862 119862 119861 119862
119860 119861 119862
119860 119860 119861 119862
119861 119862 119861 119862
119862 119862 119861 119862
119860 119861 119862
119860 119860 119862 119862
119861 119861 119861 119861
119862 119862 119862 119862
119860 119861 119862
119860 119861 119862 119861
119861 119862 119861 119862
119862 119861 119862 119861
(15)
Semigroups ⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and⟨1205907
1205908
⟩ have four elements and Cayley tables as follows
119860 119861 119862 119863
119860 119862 119863 119860 119861
119861 119860 119861 119862 119863
119862 119860 119861 119862 119863
119863 119862 119863 119860 119861
119860 119861 119862 119863
119860 119862 119860 119860 119862
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119861 119863 119863 119861
119860 119861 119862 119863
119860 119860 119862 119862 119860
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119863 119861 119861 119863
(16)
Semigroups ⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong
⟨1205904
1205906
⟩ have five elements Semigroups ⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590
0
1205902
1205905
⟩ has the followingCayley table
0 119860 119861 119862 119863 119864 119865
0 0 119860 119861 119862 119863 119864 119865
119860 119860 119860 119862 119862 119864 119864 119862
119861 119861 119863 119861 119865 119863 119863 119865
119862 119862 119864 119862 119862 119864 119864 119862
119863 119863 119863 119865 119865 119863 119863 119865
119864 119864 119864 119862 119862 119864 119864 119862
119865 119865 119863 119865 119865 119863 119863 119865
(17)
Preparing theCayley table for ⟨1205900
1205902
1205905
⟩weput1205900
= 0 1205902
=
119860 1205905
= 119861 1205907
= 119862 1205908
= 119863 12059010
= 119864 and 12059013
= 119865 This tableimmediately shows that semigroups ⟨120590
2
1205905
⟩ and ⟨1205900
1205902
1205905
⟩
have exactly two automorphisms These are identities andrestrictions of A
Semigroup ⟨1205906
1205909
⟩ has eight elements Semigroups⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ have nine ele-ments Semigroups ⟨120590
2
1205903
1205908
⟩ cong ⟨1205904
1205905
1205907
⟩ ⟨1205902
1205904
1205907
⟩ cong
⟨1205903
1205905
1205908
⟩ and ⟨1205902
1205905
1205906
⟩ have ten elements Semigroups⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩ have eleven elements In the endthe semigroup ⟨120590
3
1205904
⟩ has twelve elementsThus the semigroup ⟨120590
3
1205904
⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids
9 Viewing Semigroups Contained in M
91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +
6 Journal of Mathematics
4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590
3
1205904
⟩ fifty-seven monoidsformed by adding 120590
0
to a semigroup contained in ⟨1205903
1205904
⟩groups ⟨120590
0
⟩ ⟨1205901
⟩ and monoids ⟨1205901
1205906
⟩ = M1
and ⟨1205901
1205902
⟩ =
MThere are fifty-six types of nonisomorphic semigroups
in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590
3
1205904
⟩ twenty-six types of non-isomorphic monoids formed by adding 120590
0
to a semigroupcontained in ⟨120590
3
1205904
⟩ and alsoM1
andM Indeed adding 1205900
to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590
0
to a monoid contained in ⟨1205903
1205904
⟩ weget two new non-isomorphism types ofmonoids But adding1205900
to a group contained in ⟨1205903
1205904
⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590
2
12059010
⟩ or⟨1205902
1205906
⟩ The other two types areM1
andM
92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows
(1) ⟨1205902
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
(2) ⟨120590
2
1205903
1205905
⟩ = 1205902
1205903
1205905
1205906
12059013
(3) ⟨120590
2
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩ = 1205902
1205903
1205906
1205907
12059013
(4) ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
(5) ⟨120590
2
1205904
1205905
⟩ = 1205902
1205904
1205905
12059013
(6) ⟨120590
2
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩ = 1205902
1205904
1205906
1205907
12059013
(7) ⟨120590
2
1205905
⟩ = 1205902
1205905
1205907
1205908
12059010
12059013
(8) ⟨120590
2
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩ = 1205902
1205905
1205906
12059013
(9) ⟨120590
2
1205907
⟩ = 1205902
1205907
12059010
(10) ⟨120590
2
1205908
⟩ = 1205902
1205908
12059010
(11) ⟨120590
2
1205909
⟩ = ⟨1205902
1205906
12059013
⟩ = ⟨1205902
1205907
12059012
⟩ = ⟨1205902
1205908
12059011
⟩ = ⟨1205902
12059011
12059012
⟩ = ⟨1205902
12059011
12059013
⟩ = ⟨1205902
12059012
12059013
⟩ = 1205902
1205906
1205907
12059013
(12) ⟨120590
2
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(13) ⟨120590
2
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(14) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
(15) ⟨120590
3
⟩ = 1205903
1205907
12059011
(16) ⟨120590
3
1205904
⟩ = 1205902
1205903
12059013
(17) ⟨120590
3
1205905
⟩ = 1205903
1205905
1205907
1205909
12059011
12059013
(18) ⟨120590
3
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩ = 1205903
1205905
1205906
12059013
(19) ⟨120590
3
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(20) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(21) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
(22) ⟨120590
4
⟩ = 1205904
1205908
12059012
(23) ⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
(24) ⟨120590
4
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩ = 1205904
1205905
12059013
(25) ⟨120590
4
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
(26) ⟨120590
4
1205907
⟩ = ⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
(27) ⟨120590
4
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
(28) ⟨120590
5
1205906
⟩ = ⟨1205905
1205907
12059012
⟩ = ⟨1205905
1205908
12059011
⟩ = ⟨1205905
1205909
12059010
⟩ = ⟨1205905
12059010
12059011
⟩ = ⟨1205905
12059010
12059012
⟩ = ⟨1205905
12059011
12059012
⟩ = 1205905
1205906
12059013
(29) ⟨120590
5
1205907
⟩ = 1205905
1205907
12059013
(30) ⟨120590
5
1205908
⟩ = 1205905
1205908
12059013
(31) ⟨120590
5
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
(32) ⟨120590
5
12059011
⟩ = ⟨1205905
1205907
1205909
⟩ = 1205905
1205907
1205909
12059011
12059013
(33) ⟨120590
5
12059012
⟩ = ⟨1205905
1205908
1205909
⟩ = 1205905
1205908
1205909
12059012
12059013
(34) ⟨120590
6
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(35) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(36) ⟨120590
6
1205909
⟩ = ⟨1205906
12059013
⟩ = ⟨1205907
12059012
⟩ = ⟨1205908
12059011
⟩ = ⟨1205909
12059010
⟩ = ⟨12059011
12059012
⟩ = ⟨1205906
1205907
1205908
⟩ = ⟨1205907
1205908
1205909
⟩ =⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ = 1205906
1205907
12059013
(37) ⟨120590
7
1205908
⟩ = 1205907
1205908
12059010
12059013
(38) ⟨120590
7
1205909
⟩ = 1205907
1205909
12059011
12059013
(39) ⟨120590
7
12059010
⟩ = 1205907
12059010
(40) ⟨120590
7
12059013
⟩ = 1205907
12059013
(41) ⟨120590
8
1205909
⟩ = 1205908
1205909
12059012
12059013
(42) ⟨120590
8
12059010
⟩ = 1205908
12059010
(43) ⟨120590
8
12059013
⟩ = 1205908
12059013
93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on
(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups
(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590
0
added to 6 one-element groups and ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205908
12059010
⟩ cong ⟨1205907
12059013
⟩(iii) The monoid M contains 12 three-element monoids
with 5 isomorphism types These are 1205900
addedto 4 two-element groups and also ⟨120590
0
1205902
12059010
⟩ cong
⟨1205900
1205905
12059013
⟩ ⟨1205900
1205907
12059010
⟩ cong ⟨1205900
1205908
12059013
⟩⟨1205900
1205908
12059010
⟩ cong ⟨1205900
1205907
12059013
⟩ and ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩(iv) The monoidM contains 6 three-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong
⟨1205904
⟩
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
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2 Journal of Mathematics
cancellation rules (relations) motivated by some topologicalspaces
2 Cancellation Rules
A map 119891 119875(119883) rarr 119875(119883) is called
(i) increasing if 119860 sube 119861 implies 119891(119860) sube 119891(119861)(ii) decreasing if 119860 sube 119861 implies 119891(119861) sube 119891(119860)(iii) an involution if the composition 119891 ∘ 119891 is the identity(iv) an idempotent if 119891 ∘ 119891 = 119891
Assume that 119860 997891rarr 1205900
(119860) is the identity 119860 997891rarr 1205901
(119860) isa decreasing involution and 119860 997891rarr 120590
2
(119860) is an increasingidempotent map Other operations 120590
119894
let be compositions of1205901
and 1205902
as it was with the kuratowski operationsWe get thefollowing cancellation rules
Lemma 1 If 119861 sube 1205902
(119861) then
1205902
∘ 12059012
= 1205906
1205902
∘ 12059013
= 1205907
(2)
Proof For clarity of this proof use designations 1205901
(119860) = 119860119888
and 1205902
(119860) = 119860minus Thus we shall prove
119860minus119888minus119888minus119888minus
= 119860minus119888minus
119860119888minus119888minus119888minus119888minus
= 119860119888minus119888minus
(3)
We start with 119860minus119888minus119888 sube 119860minus119888minus119888minus substituting 119861 = 119860minus119888minus119888 in 119861 sube119861minus This corresponds to 119860minus119888minus119888minus119888 sube 119860minus119888minus119888119888 = 119860minus119888minus since 120590
1
isa decreasing involution Hence 119860minus119888minus119888minus119888minus sube 119860minus119888minus since 120590
2
anincreasing idempotent
Since 1205901
is decreasing 1205902
is increasing and 119861 sube 119861minus wehave
119861minus119888
sube 119861119888
sube 119861119888minus
(4)
Thus119860minus119888minus119888 sube 119860minus119888119888minus = 119860minus if we put119860minus119888 = 119861 Again using that1205902
is increasing and 1205901
is decreasing we obtain 119860minus119888minus119888minus sube 119860minusand then 119860minus119888 sube 119860minus119888minus119888minus119888 Finally we get 119860minus119888minus sube 119860minus119888minus119888minus119888minus
With 119860119888 in the place of 119860 in the rule 119860minus119888minus = 119860minus119888minus119888minus119888minus weget the second rule
In academic textbooks of general topology for example[2 Problem 171] one can find a hint suggested to prove theabove cancellation rules Students go like this steps 119860 sube 119860
minus
and 119860119888 sube 119860119888minus lead to 119860119888minus119888 sube 119860 the special case 119860minus119888minus119888minus119888 sube
119860minus119888minus (of119860119888minus119888 sube 119860) leads to119860minus119888minus119888minus119888minus sube 119860minus119888minus steps119860minus119888minus119888 sube 119860minus
and 119860minus119888minus119888minus sube 119860minus lead to 119860minus119888minus sube 119860minus119888minus119888minus119888minus at the end use thelast step of the proof of Lemma 1 Note that the above proofsdo not use the axioms of topology as follows
(i) 0 = 0minus(ii) (119860 cup 119861)minus = 119860minus cup 119861minus
In the literature there are articles in which Kuratowski oper-ations are replaced by some other mappings For exampleKoenen [6] considered linear spaces and put 120590
2
(119860) to bethe convex hull of 119860 In fact Shum [7] considered 120590
2
as theclosure due to the algebraic operations Add to this that theseoperations can be applied to so-called Frechet (V) spaceswhich were considered in the book [8 pages 3ndash37]
3 The Monoid M
LetM be the monoid consisting of all Kuratowski operationsthat is there are assumed cancellation rules 120590
6
= 1205902
∘12059012
and1205907
= 1205902
∘ 12059013
Fill in the Cayley table for M where the rowand column marked by the identity are omitted Similarly asin [9] the factor that labels the row comes first and that thefactor that labels the column is second For example 120590
119894
∘ 120590119896
isin the row marked by 120590
119894
and the column marked by 120590119896
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
12059010
12059011
12059012
12059013
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
12059012
12059013
12059010
12059011
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205903
1205902
1205906
1205907
1205902
1205903
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
1205905
1205904
1205908
1205909
1205904
1205905
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
1205907
1205906
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
1205909
1205908
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
12059010
12059011
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
12059011
12059010
1205906
1205907
12059010
12059011
12059010
12059011
1205906
1205907
1205906
1205907
12059010
12059011
12059012
12059013
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
12059013
12059012
1205908
1205909
12059012
12059013
12059012
12059013
1205908
1205909
1205908
1205909
12059012
12059013
(5)
It turns out that the above table allows us to describeall semigroups contained inM using pencil-and-paper tech-niques only The argument will be by a systematic study ofpossibilities Preparing the list of all semigroups consisting ofKuratowski operations we used following principles
(i) minimal collection of generators is written using⟨119860 119861 119885⟩ where letters denote generators
(ii) when a semigroup has a few minimal collections ofgenerators then its name is the first collection in thedictionary order
(iii) all minimal collections of generators are written withthe exception of some containing 120590
0
(iv) we leave to the readers verification that our list is
complete sometimes we add hints
4 Semigroups with 1205901
Observe that each semigroup which contains 1205900
is a monoidSince120590
1
∘ 1205901
= 1205900
a semigroupwhich contains1205901
is amonoidtoo
Theorem 2 There are three monoids containing 1205901
(1) ⟨1205901
⟩ = 1205900
1205901
(2) M = ⟨120590
1
120590119894
⟩ = 1205900
1205901
12059013
where 119894 isin 2 3 4 5(3) letM
1
= ⟨1205901
1205906
⟩ If 119895 isin 6 7 13 then
M1
= ⟨1205901
120590119895
⟩ = 1205900
1205901
cup 1205906
1205907
12059013
(6)
Proof The equality ⟨1205901
⟩ = 1205900
1205901
is obviousSince 120590
2
= 1205903
∘ 1205901
= 1205901
∘ 1205904
= 1205901
∘ 1205905
∘ 1205901
we have⟨1205901
1205902
⟩ = ⟨1205901
1205903
⟩ = ⟨1205901
1205904
⟩ = ⟨1205901
1205905
⟩ = MIf 119895 isin 6 7 13 then any composition 120590
119895
∘ 120590119894
or 120590119896
∘ 120590119895
belongs to M1
and so ⟨1205901
120590119895
⟩ sube M1
We have 1205907
= 1205906
∘ 1205901
Journal of Mathematics 3
1205908
= 1205901
∘ 1205906
1205909
= 1205901
∘ 1205907
12059010
= 1205906
∘ 1205906
12059011
= 1205906
∘ 1205907
12059012
= 1205908
∘ 1205906
and 12059013
= 1205908
∘ 1205907
and so M1
= ⟨1205901
1205906
⟩ =
1205900
1205901
cup 1205906
1205907
12059013
Since 120590
6
= 1205907
∘ 1205901
= 1205901
∘ 1205908
= 1205901
∘ 1205909
∘ 1205901
= 12059010
∘ 1205901
∘ 12059010
=
12059011
∘ 12059011
∘ 1205901
= 1205901
∘ 12059012
∘ 12059012
and 1205906
= 1205901
∘ 12059013
∘ 1205901
∘ 12059013
∘ 1205901
we have ⟨120590
1
120590119895
⟩ = M1
for 119895 isin 6 7 13
Consider the permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
12059010
12059011
12059012
12059013
1205900
1205901
1205905
1205904
1205903
1205902
1205909
1205908
1205907
1205906
12059013
12059012
12059011
12059010
)
(7)
It determines an automorphism A M rarr M
Theorem 3 The identity and A are the only automorphismsofM
Proof Delete rows and columns marked by 1205901
in the Cayleytable for M Then check that the operation 120590
3
is in the rowor the column marked by 120590
3
only Also the operation 1205904
isin the row or the column marked by 120590
4
only Therefore thesemigroup
⟨1205903
1205904
⟩ = 1205902
1205903
12059013
(8)
has a unique minimal set of generators 1205903
1205904
The reader isleft to check this with the Cayley table forM
Suppose G is an automorphism of M By Theorem 2 Gtransforms the set 120590
2
1205903
1205904
1205905
onto itself However 1205902
and1205905
are idempotents but 1205903
and 1205904
are not idempotents Sothere are two possibilities G(120590
3
) = 1205903
and G(1205904
) = 1205904
whichimplies that G is the identity G(120590
3
) = 1205904
and G(1205904
) = 1205903
which implies G = A The reader is left to check this with theCayley table forM We offer hints 120590
2
= 1205903
∘ 1205904
1205905
= 1205904
∘ 1205903
1205906
= 1205903
∘ 1205902
1205907
= 1205903
∘ 1205903
1205908
= 1205904
∘ 1205904
1205909
= 1205904
∘ 1205905
12059010
= 1205906
∘ 1205906
12059011
= 1205906
∘ 1205907
12059012
= 1205908
∘ 1205906
and 12059013
= 1205908
∘ 1205907
to verify the details of this proof
5 The Monoid of All Idempotents
The set 1205900
1205902
1205905
1205907
1205908
12059010
12059013
consists of all squares inMThese squares are idempotents and lie on the main diagonalin the Cayley table forM They constitute a monoid and
1205900
1205902
1205905
1205907
1205908
12059010
12059013
= ⟨1205900
1205902
1205905
⟩ (9)
The permutation
(1205900
1205902
1205905
1205907
1205908
12059010
12059013
1205900
1205902
1205905
1205908
1205907
12059010
12059013
) (10)
determines the bijection I ⟨1205900
1205902
1205905
⟩ rarr ⟨1205900
1205902
1205905
⟩ suchthat
I (120572 ∘ 120573) = I (120573) ∘ I (120572) (11)
for any 120572 120573 isin ⟨1205900
1205902
1205905
⟩ To verify this apply equalities1205902
∘ 1205905
= 1205907
1205905
∘ 1205902
= 1205908
1205902
∘ 1205905
∘ 1205902
= 12059010
and1205905
∘ 1205902
∘ 1205905
= 12059013
Any bijection I 119866 rarr 119867 having propertyI(120572∘120573) = I(120573)∘I(120572) transposesCayley tables for semigroups119866
and119867 Several semigroups contained inM have this propertyWe leave the reader to verify this
We shall classify all semigroups contained in the semi-group ⟨120590
2
1205905
⟩ Every such semigroup can be extended toa monoid by attaching 120590
0
to it This gives a completeclassification of all semigroups in ⟨120590
0
1205902
1205905
⟩The semigroup ⟨120590
2
1205905
⟩ contains six groups with exactlyone element
Semigroups ⟨1205902
12059010
⟩ = 1205902
12059010
and ⟨1205905
12059013
⟩ = 1205905
12059013
aremonoids Both consist of exactly two elements and are notgroups so they are isomorphic
Semigroups ⟨1205907
12059010
⟩ and ⟨1205908
12059013
⟩ are isomorphic in par-ticular A[⟨120590
7
12059010
⟩] = ⟨1205908
12059013
⟩ Also semigroups ⟨1205907
12059013
⟩
and ⟨1205908
12059010
⟩ are isomorphic by A Every of these four semi-groups has exactly two elements None of them is a monoidThey form two types of nonisomorphic semigroups becausethe bijections (120590
7
12059010
) (12059010
1205908
) and (1205907
1205908
) (12059010
12059010
) arenot isomorphisms
Semigroups ⟨1205902
1205907
⟩ = 1205902
1205907
12059010
and ⟨1205905
1205908
⟩ =
1205905
1205908
12059013
are isomorphic Also semigroups ⟨1205902
1205908
⟩ =
1205902
1205908
12059010
and ⟨1205905
1205907
⟩ = 1205905
1205907
12059013
are isomorphic Infact A[⟨120590
2
1205907
⟩] = ⟨1205905
1205908
⟩ and A[⟨1205902
1205908
⟩] = ⟨1205905
1205907
⟩None of these semigroups is a monoid They form two typesof nonisomorphic semigroups Indeed any isomorphismbetween ⟨120590
2
1205907
⟩ and ⟨1205902
1205908
⟩ must be the identity on themonoid ⟨120590
2
12059010
⟩ Therefore would have to be the restrictionof I But I restricted to ⟨120590
7
12059010
⟩ is not an isomorphismSemigroups ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
and ⟨1205905
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
are notmonoids They are isomorphic by A
The semigroup ⟨1205902
1205905
⟩ contains exactly one semigroupwith four elements ⟨120590
7
1205908
⟩ = ⟨12059010
12059013
⟩ = 1205907
1205908
12059010
12059013
which is not a monoidNote that ⟨120590
2
1205905
⟩ contains twenty different semigroupswith nine non-isomorphism types These are six isomorphicgroups with exactly one element ⟨120590
2
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong
⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ two isomorphic monoids with exactlytwo elements ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ two pairs of isomorphicsemigroups with exactly two elements ⟨120590
7
12059010
⟩ cong ⟨1205908
12059013
⟩
and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ two pairs of isomorphic semigroupswith exactly three elements ⟨120590
2
1205907
⟩ cong ⟨1205905
1205908
⟩ and ⟨1205902
1205908
⟩ cong
⟨1205905
1205907
⟩ a semigroup with exactly four elements ⟨1205907
1205908
⟩and two isomorphic semigroups with exactly five elements⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ and also ⟨1205902
1205905
⟩ Thus ⟨1205902
1205905
⟩ containstwenty different semigroups with nine isomorphism typesBut ⟨120590
0
1205902
1205905
⟩ contains forty-one different semigroups withseventeen isomorphism types Indeed adding 120590
0
to semi-groups contained in ⟨120590
2
1205905
⟩ which are not monoids weget twenty monoids with eight non-isomorphism typesAdding 120590
0
to a group contained in ⟨1205902
1205905
⟩ we get a monoidisomorphic to ⟨120590
2
12059010
⟩
6 The Semigroup Consisting of 12059061205907 120590
13
Using the Cayley table forM check thatA [1205906
1205907
12059013
] = 1205906
1205907
12059013
(12)Similarly check that the semigroup ⟨120590
6
1205909
⟩ = 1205906
1205907
12059013
can be represented as ⟨1205906
12059013
⟩ ⟨1205907
12059012
⟩ ⟨1205908
12059011
⟩
4 Journal of Mathematics
⟨1205909
12059010
⟩ or ⟨12059011
12059012
⟩ Also ⟨1205906
1205909
⟩ = ⟨1205906
1205907
1205908
⟩ =
⟨1205907
1205908
1205909
⟩ = ⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590
6
1205909
⟩ Other semigroupsincluded in ⟨120590
6
1205909
⟩ have one or two minimal collectionof generators One generator has groups ⟨120590
6
⟩ = 1205906
12059010
⟨1205909
⟩ = 1205909
12059013
⟨12059011
⟩ = 1205907
12059011
and ⟨12059012
⟩ = 1205908
12059012
Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590
7
12059010
⟩ ⟨1205907
12059013
⟩ ⟨1205908
12059010
⟩ ⟨1205908
12059013
⟩ and⟨1205907
1205908
⟩ are discussed in the previous section Contained in⟨1205906
1205909
⟩ and not previously discussed semigroups are ⟨1205906
1205907
⟩⟨1205906
1205908
⟩⟨1205907
1205909
⟩ and ⟨1205908
1205909
⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590
6
1205909
⟩ but different from thewhole One has
(i) ⟨1205906
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(ii) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(iii) ⟨120590
7
1205909
⟩ = ⟨1205909
12059011
⟩ = ⟨12059011
12059013
⟩ = 1205907
1205909
12059011
12059013
(iv) ⟨120590
8
1205909
⟩ = ⟨1205909
12059012
⟩ = ⟨12059012
12059013
⟩ = 1205908
1205909
12059012
12059013
Proposition 4 Semigroups ⟨1205906
1205907
⟩ and ⟨1205908
1205909
⟩ are isomor-phic and also semigroups ⟨120590
6
1205908
⟩ and ⟨1205907
1205909
⟩ are isomorphicbut semigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ are not isomorphic
Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906
1205907
⟩ rarr ⟨1205906
1205908
⟩ is an isomorphism Thus 119869[⟨1205907
12059010
⟩] =
⟨1205908
12059010
⟩ Given 119869(1205907
) = 1205908
we get 12059010
= 119869(12059010
) = 119869(1205907
∘
12059010
) = 1205908
∘ 12059010
= 1205908
But 119869(1205907
) = 12059010
implies 12059010
= 119869(1205907
) =
119869(12059010
∘ 1205907
) = 1205908
∘ 12059010
= 1205908
Both possibilities lead to acontradiction
So ⟨1205906
1205909
⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590
7
⟩ cong ⟨1205908
⟩ cong
⟨12059010
⟩ cong ⟨12059013
⟩ four isomorphic groups with exactly twoelements ⟨120590
6
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ and five semigroupswith exactly four elements ⟨120590
6
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong
⟨1205907
1205909
⟩ ⟨1205907
1205908
⟩ and also ⟨1205906
1205909
⟩
7 Remaining Semigroups in ⟨1205903 1205904⟩
We have yet to discuss semigroups included in ⟨1205903
1205904
⟩ notincluded in ⟨120590
2
1205905
⟩ and containing at least one of Kuratowskioperation 120590
2
1205903
1205904
or 1205905
It will be discussed up to theisomorphism A Obviously ⟨120590
2
⟩ = 1205902
and ⟨1205905
⟩ = 1205905
aregroups
71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590
2
1205906
⟩ = 1205902
1205906
12059010
and ⟨1205902
12059010
⟩ = 1205902
12059010
havedifferent numbers of elements Also ⟨120590
2
1205906
⟩ is isomorphic to⟨1205900
1205906
⟩ Nonisomorphic semigroups ⟨1205902
1205907
⟩ and ⟨1205902
1205908
⟩ arediscussed above The following three semigroups
(i) ⟨1205902
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(ii) ⟨1205902
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(iii) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ or ⟨1205907
1205908
⟩ This isimpossible by Proposition 4 and because ⟨120590
7
1205908
⟩ consists ofidempotents but 120590
6
is not an idempotent The nine-elementsemigroup on the set 120590
2
1205906
1205907
12059013
is represented as⟨1205902
1205909
⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590
2
12059011
⟩ ⟨1205902
12059012
⟩ ⟨1205902
12059013
⟩ and ⟨1205902
1205909
⟩Using A we have described eight semigroupsmdasheach
one isomorphic to a semigroup previously discussedmdashwhichcontains 120590
5
and elements (at least one) of ⟨1205906
1205909
⟩ Collec-tions of generators ⟨120590
2
1205906
12059013
⟩ ⟨1205902
1205907
12059012
⟩ ⟨1205902
1205908
12059011
⟩⟨1205902
12059011
12059012
⟩ ⟨1205902
12059011
12059013
⟩ and ⟨1205902
12059012
12059013
⟩ are minimalin ⟨120590
2
1205909
⟩ Also collections of generators ⟨1205905
1205907
12059012
⟩⟨1205905
1205908
12059011
⟩ ⟨1205905
1205909
12059010
⟩ ⟨1205905
12059010
12059011
⟩ ⟨1205905
12059010
12059012
⟩ and⟨1205905
12059011
12059012
⟩ are minimal in ⟨1205905
1205906
⟩
72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590
3
⟩ = 1205903
1205907
12059011
and ⟨1205904
⟩ =
1205904
1205908
12059012
are isomorphic by A They are not monoids Thesemigroup ⟨120590
3
⟩ can be extended using elements of ⟨1205906
1205909
⟩in three following ways
(i) ⟨1205903
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(ii) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(iii) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
Semigroups ⟨1205903
1205906
⟩ and ⟨1205903
1205909
⟩ are not isomorphic Indeedsuppose 119869 ⟨120590
3
1205906
⟩ rarr ⟨1205903
1205909
⟩ is an isomorphismThus 119869 isthe identity on ⟨120590
3
⟩ 119869(1205906
) = 1205909
and 119869(12059010
) = 12059013
This givesa contradiction since 120590
3
∘ 1205906
= 12059010
and 1205903
∘ 1205909
= 1205907
Neither ⟨120590
3
1205906
⟩ nor ⟨1205903
1205909
⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590
2
1205909
⟩ is not isomorphic to ⟨1205903
1205908
⟩since ⟨120590
2
1205909
⟩ has a unique pair of generators and ⟨1205903
1205908
⟩ =
⟨1205903
12059012
⟩Using A we getmdashisomorphic to previously discussed
onesmdashsemigroups ⟨1205904
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
⟨1205904
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
and ⟨1205904
1205907
⟩ =
⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
There exist minimal collec-tions of generators such as follows
⟨1205903
1205908
⟩ = ⟨1205903
1205906
1205909
⟩ = ⟨1205903
1205906
12059013
⟩
= ⟨1205903
1205909
12059010
⟩ = ⟨1205903
12059010
12059013
⟩
⟨1205904
1205907
⟩ = ⟨1205904
1205906
1205909
⟩ = ⟨1205904
1205909
12059010
⟩
= ⟨1205904
1205906
12059013
⟩ = ⟨1205904
12059010
12059013
⟩
(13)
73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590
2
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
and⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
= A[⟨1205902
1205903
⟩] andalso ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
and ⟨1205903
1205905
⟩ =
1205903
1205905
1205907
1205909
12059011
12059013
= A[⟨1205902
1205904
⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types
Journal of Mathematics 5
Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590
2
1205905
⟩ has 6 elements which are idempotentsIn ⟨1205903
1205904
⟩ there are seven semigroups which have threegenerators and have not two generators These are
(1) ⟨1205902
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩
(2) ⟨1205904
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩
(3) ⟨1205902
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩
(4) ⟨1205903
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩
(5) ⟨1205902
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩
(6) ⟨1205902
1205903
1205905
⟩
(7) ⟨1205902
1205904
1205905
⟩
8 Semigroups which Are Contained in ⟨12059031205904⟩
Groups ⟨1205902
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong ⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ have oneelement and are isomorphic
Groups ⟨1205906
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ monoids ⟨1205902
12059010
⟩ cong
⟨1205905
12059013
⟩ and also semigroups ⟨1205907
12059010
⟩ cong ⟨1205908
12059013
⟩ and⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ have two elements and Cayley tables asfollows
119860 119861
119860 119861 119860
119861 119860 119861
119860 119861
119860 119860 119861
119861 119861 119860
119860 119861
119860 119860 119861
119861 119860 119861
119860 119861
119860 119860 119860
119861 119861 119861
(14)
Monoids ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩ and also semigroups⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong ⟨1205904
⟩ havethree elements and Cayley tables as follows
119860 119861 119862
119860 119860 119861 119862
119861 119861 119862 119861
119862 119862 119861 119862
119860 119861 119862
119860 119860 119861 119862
119861 119862 119861 119862
119862 119862 119861 119862
119860 119861 119862
119860 119860 119862 119862
119861 119861 119861 119861
119862 119862 119862 119862
119860 119861 119862
119860 119861 119862 119861
119861 119862 119861 119862
119862 119861 119862 119861
(15)
Semigroups ⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and⟨1205907
1205908
⟩ have four elements and Cayley tables as follows
119860 119861 119862 119863
119860 119862 119863 119860 119861
119861 119860 119861 119862 119863
119862 119860 119861 119862 119863
119863 119862 119863 119860 119861
119860 119861 119862 119863
119860 119862 119860 119860 119862
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119861 119863 119863 119861
119860 119861 119862 119863
119860 119860 119862 119862 119860
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119863 119861 119861 119863
(16)
Semigroups ⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong
⟨1205904
1205906
⟩ have five elements Semigroups ⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590
0
1205902
1205905
⟩ has the followingCayley table
0 119860 119861 119862 119863 119864 119865
0 0 119860 119861 119862 119863 119864 119865
119860 119860 119860 119862 119862 119864 119864 119862
119861 119861 119863 119861 119865 119863 119863 119865
119862 119862 119864 119862 119862 119864 119864 119862
119863 119863 119863 119865 119865 119863 119863 119865
119864 119864 119864 119862 119862 119864 119864 119862
119865 119865 119863 119865 119865 119863 119863 119865
(17)
Preparing theCayley table for ⟨1205900
1205902
1205905
⟩weput1205900
= 0 1205902
=
119860 1205905
= 119861 1205907
= 119862 1205908
= 119863 12059010
= 119864 and 12059013
= 119865 This tableimmediately shows that semigroups ⟨120590
2
1205905
⟩ and ⟨1205900
1205902
1205905
⟩
have exactly two automorphisms These are identities andrestrictions of A
Semigroup ⟨1205906
1205909
⟩ has eight elements Semigroups⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ have nine ele-ments Semigroups ⟨120590
2
1205903
1205908
⟩ cong ⟨1205904
1205905
1205907
⟩ ⟨1205902
1205904
1205907
⟩ cong
⟨1205903
1205905
1205908
⟩ and ⟨1205902
1205905
1205906
⟩ have ten elements Semigroups⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩ have eleven elements In the endthe semigroup ⟨120590
3
1205904
⟩ has twelve elementsThus the semigroup ⟨120590
3
1205904
⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids
9 Viewing Semigroups Contained in M
91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +
6 Journal of Mathematics
4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590
3
1205904
⟩ fifty-seven monoidsformed by adding 120590
0
to a semigroup contained in ⟨1205903
1205904
⟩groups ⟨120590
0
⟩ ⟨1205901
⟩ and monoids ⟨1205901
1205906
⟩ = M1
and ⟨1205901
1205902
⟩ =
MThere are fifty-six types of nonisomorphic semigroups
in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590
3
1205904
⟩ twenty-six types of non-isomorphic monoids formed by adding 120590
0
to a semigroupcontained in ⟨120590
3
1205904
⟩ and alsoM1
andM Indeed adding 1205900
to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590
0
to a monoid contained in ⟨1205903
1205904
⟩ weget two new non-isomorphism types ofmonoids But adding1205900
to a group contained in ⟨1205903
1205904
⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590
2
12059010
⟩ or⟨1205902
1205906
⟩ The other two types areM1
andM
92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows
(1) ⟨1205902
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
(2) ⟨120590
2
1205903
1205905
⟩ = 1205902
1205903
1205905
1205906
12059013
(3) ⟨120590
2
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩ = 1205902
1205903
1205906
1205907
12059013
(4) ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
(5) ⟨120590
2
1205904
1205905
⟩ = 1205902
1205904
1205905
12059013
(6) ⟨120590
2
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩ = 1205902
1205904
1205906
1205907
12059013
(7) ⟨120590
2
1205905
⟩ = 1205902
1205905
1205907
1205908
12059010
12059013
(8) ⟨120590
2
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩ = 1205902
1205905
1205906
12059013
(9) ⟨120590
2
1205907
⟩ = 1205902
1205907
12059010
(10) ⟨120590
2
1205908
⟩ = 1205902
1205908
12059010
(11) ⟨120590
2
1205909
⟩ = ⟨1205902
1205906
12059013
⟩ = ⟨1205902
1205907
12059012
⟩ = ⟨1205902
1205908
12059011
⟩ = ⟨1205902
12059011
12059012
⟩ = ⟨1205902
12059011
12059013
⟩ = ⟨1205902
12059012
12059013
⟩ = 1205902
1205906
1205907
12059013
(12) ⟨120590
2
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(13) ⟨120590
2
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(14) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
(15) ⟨120590
3
⟩ = 1205903
1205907
12059011
(16) ⟨120590
3
1205904
⟩ = 1205902
1205903
12059013
(17) ⟨120590
3
1205905
⟩ = 1205903
1205905
1205907
1205909
12059011
12059013
(18) ⟨120590
3
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩ = 1205903
1205905
1205906
12059013
(19) ⟨120590
3
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(20) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(21) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
(22) ⟨120590
4
⟩ = 1205904
1205908
12059012
(23) ⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
(24) ⟨120590
4
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩ = 1205904
1205905
12059013
(25) ⟨120590
4
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
(26) ⟨120590
4
1205907
⟩ = ⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
(27) ⟨120590
4
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
(28) ⟨120590
5
1205906
⟩ = ⟨1205905
1205907
12059012
⟩ = ⟨1205905
1205908
12059011
⟩ = ⟨1205905
1205909
12059010
⟩ = ⟨1205905
12059010
12059011
⟩ = ⟨1205905
12059010
12059012
⟩ = ⟨1205905
12059011
12059012
⟩ = 1205905
1205906
12059013
(29) ⟨120590
5
1205907
⟩ = 1205905
1205907
12059013
(30) ⟨120590
5
1205908
⟩ = 1205905
1205908
12059013
(31) ⟨120590
5
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
(32) ⟨120590
5
12059011
⟩ = ⟨1205905
1205907
1205909
⟩ = 1205905
1205907
1205909
12059011
12059013
(33) ⟨120590
5
12059012
⟩ = ⟨1205905
1205908
1205909
⟩ = 1205905
1205908
1205909
12059012
12059013
(34) ⟨120590
6
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(35) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(36) ⟨120590
6
1205909
⟩ = ⟨1205906
12059013
⟩ = ⟨1205907
12059012
⟩ = ⟨1205908
12059011
⟩ = ⟨1205909
12059010
⟩ = ⟨12059011
12059012
⟩ = ⟨1205906
1205907
1205908
⟩ = ⟨1205907
1205908
1205909
⟩ =⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ = 1205906
1205907
12059013
(37) ⟨120590
7
1205908
⟩ = 1205907
1205908
12059010
12059013
(38) ⟨120590
7
1205909
⟩ = 1205907
1205909
12059011
12059013
(39) ⟨120590
7
12059010
⟩ = 1205907
12059010
(40) ⟨120590
7
12059013
⟩ = 1205907
12059013
(41) ⟨120590
8
1205909
⟩ = 1205908
1205909
12059012
12059013
(42) ⟨120590
8
12059010
⟩ = 1205908
12059010
(43) ⟨120590
8
12059013
⟩ = 1205908
12059013
93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on
(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups
(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590
0
added to 6 one-element groups and ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205908
12059010
⟩ cong ⟨1205907
12059013
⟩(iii) The monoid M contains 12 three-element monoids
with 5 isomorphism types These are 1205900
addedto 4 two-element groups and also ⟨120590
0
1205902
12059010
⟩ cong
⟨1205900
1205905
12059013
⟩ ⟨1205900
1205907
12059010
⟩ cong ⟨1205900
1205908
12059013
⟩⟨1205900
1205908
12059010
⟩ cong ⟨1205900
1205907
12059013
⟩ and ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩(iv) The monoidM contains 6 three-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong
⟨1205904
⟩
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
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Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
1205908
= 1205901
∘ 1205906
1205909
= 1205901
∘ 1205907
12059010
= 1205906
∘ 1205906
12059011
= 1205906
∘ 1205907
12059012
= 1205908
∘ 1205906
and 12059013
= 1205908
∘ 1205907
and so M1
= ⟨1205901
1205906
⟩ =
1205900
1205901
cup 1205906
1205907
12059013
Since 120590
6
= 1205907
∘ 1205901
= 1205901
∘ 1205908
= 1205901
∘ 1205909
∘ 1205901
= 12059010
∘ 1205901
∘ 12059010
=
12059011
∘ 12059011
∘ 1205901
= 1205901
∘ 12059012
∘ 12059012
and 1205906
= 1205901
∘ 12059013
∘ 1205901
∘ 12059013
∘ 1205901
we have ⟨120590
1
120590119895
⟩ = M1
for 119895 isin 6 7 13
Consider the permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
12059010
12059011
12059012
12059013
1205900
1205901
1205905
1205904
1205903
1205902
1205909
1205908
1205907
1205906
12059013
12059012
12059011
12059010
)
(7)
It determines an automorphism A M rarr M
Theorem 3 The identity and A are the only automorphismsofM
Proof Delete rows and columns marked by 1205901
in the Cayleytable for M Then check that the operation 120590
3
is in the rowor the column marked by 120590
3
only Also the operation 1205904
isin the row or the column marked by 120590
4
only Therefore thesemigroup
⟨1205903
1205904
⟩ = 1205902
1205903
12059013
(8)
has a unique minimal set of generators 1205903
1205904
The reader isleft to check this with the Cayley table forM
Suppose G is an automorphism of M By Theorem 2 Gtransforms the set 120590
2
1205903
1205904
1205905
onto itself However 1205902
and1205905
are idempotents but 1205903
and 1205904
are not idempotents Sothere are two possibilities G(120590
3
) = 1205903
and G(1205904
) = 1205904
whichimplies that G is the identity G(120590
3
) = 1205904
and G(1205904
) = 1205903
which implies G = A The reader is left to check this with theCayley table forM We offer hints 120590
2
= 1205903
∘ 1205904
1205905
= 1205904
∘ 1205903
1205906
= 1205903
∘ 1205902
1205907
= 1205903
∘ 1205903
1205908
= 1205904
∘ 1205904
1205909
= 1205904
∘ 1205905
12059010
= 1205906
∘ 1205906
12059011
= 1205906
∘ 1205907
12059012
= 1205908
∘ 1205906
and 12059013
= 1205908
∘ 1205907
to verify the details of this proof
5 The Monoid of All Idempotents
The set 1205900
1205902
1205905
1205907
1205908
12059010
12059013
consists of all squares inMThese squares are idempotents and lie on the main diagonalin the Cayley table forM They constitute a monoid and
1205900
1205902
1205905
1205907
1205908
12059010
12059013
= ⟨1205900
1205902
1205905
⟩ (9)
The permutation
(1205900
1205902
1205905
1205907
1205908
12059010
12059013
1205900
1205902
1205905
1205908
1205907
12059010
12059013
) (10)
determines the bijection I ⟨1205900
1205902
1205905
⟩ rarr ⟨1205900
1205902
1205905
⟩ suchthat
I (120572 ∘ 120573) = I (120573) ∘ I (120572) (11)
for any 120572 120573 isin ⟨1205900
1205902
1205905
⟩ To verify this apply equalities1205902
∘ 1205905
= 1205907
1205905
∘ 1205902
= 1205908
1205902
∘ 1205905
∘ 1205902
= 12059010
and1205905
∘ 1205902
∘ 1205905
= 12059013
Any bijection I 119866 rarr 119867 having propertyI(120572∘120573) = I(120573)∘I(120572) transposesCayley tables for semigroups119866
and119867 Several semigroups contained inM have this propertyWe leave the reader to verify this
We shall classify all semigroups contained in the semi-group ⟨120590
2
1205905
⟩ Every such semigroup can be extended toa monoid by attaching 120590
0
to it This gives a completeclassification of all semigroups in ⟨120590
0
1205902
1205905
⟩The semigroup ⟨120590
2
1205905
⟩ contains six groups with exactlyone element
Semigroups ⟨1205902
12059010
⟩ = 1205902
12059010
and ⟨1205905
12059013
⟩ = 1205905
12059013
aremonoids Both consist of exactly two elements and are notgroups so they are isomorphic
Semigroups ⟨1205907
12059010
⟩ and ⟨1205908
12059013
⟩ are isomorphic in par-ticular A[⟨120590
7
12059010
⟩] = ⟨1205908
12059013
⟩ Also semigroups ⟨1205907
12059013
⟩
and ⟨1205908
12059010
⟩ are isomorphic by A Every of these four semi-groups has exactly two elements None of them is a monoidThey form two types of nonisomorphic semigroups becausethe bijections (120590
7
12059010
) (12059010
1205908
) and (1205907
1205908
) (12059010
12059010
) arenot isomorphisms
Semigroups ⟨1205902
1205907
⟩ = 1205902
1205907
12059010
and ⟨1205905
1205908
⟩ =
1205905
1205908
12059013
are isomorphic Also semigroups ⟨1205902
1205908
⟩ =
1205902
1205908
12059010
and ⟨1205905
1205907
⟩ = 1205905
1205907
12059013
are isomorphic Infact A[⟨120590
2
1205907
⟩] = ⟨1205905
1205908
⟩ and A[⟨1205902
1205908
⟩] = ⟨1205905
1205907
⟩None of these semigroups is a monoid They form two typesof nonisomorphic semigroups Indeed any isomorphismbetween ⟨120590
2
1205907
⟩ and ⟨1205902
1205908
⟩ must be the identity on themonoid ⟨120590
2
12059010
⟩ Therefore would have to be the restrictionof I But I restricted to ⟨120590
7
12059010
⟩ is not an isomorphismSemigroups ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
and ⟨1205905
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
are notmonoids They are isomorphic by A
The semigroup ⟨1205902
1205905
⟩ contains exactly one semigroupwith four elements ⟨120590
7
1205908
⟩ = ⟨12059010
12059013
⟩ = 1205907
1205908
12059010
12059013
which is not a monoidNote that ⟨120590
2
1205905
⟩ contains twenty different semigroupswith nine non-isomorphism types These are six isomorphicgroups with exactly one element ⟨120590
2
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong
⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ two isomorphic monoids with exactlytwo elements ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ two pairs of isomorphicsemigroups with exactly two elements ⟨120590
7
12059010
⟩ cong ⟨1205908
12059013
⟩
and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ two pairs of isomorphic semigroupswith exactly three elements ⟨120590
2
1205907
⟩ cong ⟨1205905
1205908
⟩ and ⟨1205902
1205908
⟩ cong
⟨1205905
1205907
⟩ a semigroup with exactly four elements ⟨1205907
1205908
⟩and two isomorphic semigroups with exactly five elements⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ and also ⟨1205902
1205905
⟩ Thus ⟨1205902
1205905
⟩ containstwenty different semigroups with nine isomorphism typesBut ⟨120590
0
1205902
1205905
⟩ contains forty-one different semigroups withseventeen isomorphism types Indeed adding 120590
0
to semi-groups contained in ⟨120590
2
1205905
⟩ which are not monoids weget twenty monoids with eight non-isomorphism typesAdding 120590
0
to a group contained in ⟨1205902
1205905
⟩ we get a monoidisomorphic to ⟨120590
2
12059010
⟩
6 The Semigroup Consisting of 12059061205907 120590
13
Using the Cayley table forM check thatA [1205906
1205907
12059013
] = 1205906
1205907
12059013
(12)Similarly check that the semigroup ⟨120590
6
1205909
⟩ = 1205906
1205907
12059013
can be represented as ⟨1205906
12059013
⟩ ⟨1205907
12059012
⟩ ⟨1205908
12059011
⟩
4 Journal of Mathematics
⟨1205909
12059010
⟩ or ⟨12059011
12059012
⟩ Also ⟨1205906
1205909
⟩ = ⟨1205906
1205907
1205908
⟩ =
⟨1205907
1205908
1205909
⟩ = ⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590
6
1205909
⟩ Other semigroupsincluded in ⟨120590
6
1205909
⟩ have one or two minimal collectionof generators One generator has groups ⟨120590
6
⟩ = 1205906
12059010
⟨1205909
⟩ = 1205909
12059013
⟨12059011
⟩ = 1205907
12059011
and ⟨12059012
⟩ = 1205908
12059012
Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590
7
12059010
⟩ ⟨1205907
12059013
⟩ ⟨1205908
12059010
⟩ ⟨1205908
12059013
⟩ and⟨1205907
1205908
⟩ are discussed in the previous section Contained in⟨1205906
1205909
⟩ and not previously discussed semigroups are ⟨1205906
1205907
⟩⟨1205906
1205908
⟩⟨1205907
1205909
⟩ and ⟨1205908
1205909
⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590
6
1205909
⟩ but different from thewhole One has
(i) ⟨1205906
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(ii) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(iii) ⟨120590
7
1205909
⟩ = ⟨1205909
12059011
⟩ = ⟨12059011
12059013
⟩ = 1205907
1205909
12059011
12059013
(iv) ⟨120590
8
1205909
⟩ = ⟨1205909
12059012
⟩ = ⟨12059012
12059013
⟩ = 1205908
1205909
12059012
12059013
Proposition 4 Semigroups ⟨1205906
1205907
⟩ and ⟨1205908
1205909
⟩ are isomor-phic and also semigroups ⟨120590
6
1205908
⟩ and ⟨1205907
1205909
⟩ are isomorphicbut semigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ are not isomorphic
Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906
1205907
⟩ rarr ⟨1205906
1205908
⟩ is an isomorphism Thus 119869[⟨1205907
12059010
⟩] =
⟨1205908
12059010
⟩ Given 119869(1205907
) = 1205908
we get 12059010
= 119869(12059010
) = 119869(1205907
∘
12059010
) = 1205908
∘ 12059010
= 1205908
But 119869(1205907
) = 12059010
implies 12059010
= 119869(1205907
) =
119869(12059010
∘ 1205907
) = 1205908
∘ 12059010
= 1205908
Both possibilities lead to acontradiction
So ⟨1205906
1205909
⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590
7
⟩ cong ⟨1205908
⟩ cong
⟨12059010
⟩ cong ⟨12059013
⟩ four isomorphic groups with exactly twoelements ⟨120590
6
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ and five semigroupswith exactly four elements ⟨120590
6
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong
⟨1205907
1205909
⟩ ⟨1205907
1205908
⟩ and also ⟨1205906
1205909
⟩
7 Remaining Semigroups in ⟨1205903 1205904⟩
We have yet to discuss semigroups included in ⟨1205903
1205904
⟩ notincluded in ⟨120590
2
1205905
⟩ and containing at least one of Kuratowskioperation 120590
2
1205903
1205904
or 1205905
It will be discussed up to theisomorphism A Obviously ⟨120590
2
⟩ = 1205902
and ⟨1205905
⟩ = 1205905
aregroups
71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590
2
1205906
⟩ = 1205902
1205906
12059010
and ⟨1205902
12059010
⟩ = 1205902
12059010
havedifferent numbers of elements Also ⟨120590
2
1205906
⟩ is isomorphic to⟨1205900
1205906
⟩ Nonisomorphic semigroups ⟨1205902
1205907
⟩ and ⟨1205902
1205908
⟩ arediscussed above The following three semigroups
(i) ⟨1205902
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(ii) ⟨1205902
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(iii) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ or ⟨1205907
1205908
⟩ This isimpossible by Proposition 4 and because ⟨120590
7
1205908
⟩ consists ofidempotents but 120590
6
is not an idempotent The nine-elementsemigroup on the set 120590
2
1205906
1205907
12059013
is represented as⟨1205902
1205909
⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590
2
12059011
⟩ ⟨1205902
12059012
⟩ ⟨1205902
12059013
⟩ and ⟨1205902
1205909
⟩Using A we have described eight semigroupsmdasheach
one isomorphic to a semigroup previously discussedmdashwhichcontains 120590
5
and elements (at least one) of ⟨1205906
1205909
⟩ Collec-tions of generators ⟨120590
2
1205906
12059013
⟩ ⟨1205902
1205907
12059012
⟩ ⟨1205902
1205908
12059011
⟩⟨1205902
12059011
12059012
⟩ ⟨1205902
12059011
12059013
⟩ and ⟨1205902
12059012
12059013
⟩ are minimalin ⟨120590
2
1205909
⟩ Also collections of generators ⟨1205905
1205907
12059012
⟩⟨1205905
1205908
12059011
⟩ ⟨1205905
1205909
12059010
⟩ ⟨1205905
12059010
12059011
⟩ ⟨1205905
12059010
12059012
⟩ and⟨1205905
12059011
12059012
⟩ are minimal in ⟨1205905
1205906
⟩
72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590
3
⟩ = 1205903
1205907
12059011
and ⟨1205904
⟩ =
1205904
1205908
12059012
are isomorphic by A They are not monoids Thesemigroup ⟨120590
3
⟩ can be extended using elements of ⟨1205906
1205909
⟩in three following ways
(i) ⟨1205903
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(ii) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(iii) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
Semigroups ⟨1205903
1205906
⟩ and ⟨1205903
1205909
⟩ are not isomorphic Indeedsuppose 119869 ⟨120590
3
1205906
⟩ rarr ⟨1205903
1205909
⟩ is an isomorphismThus 119869 isthe identity on ⟨120590
3
⟩ 119869(1205906
) = 1205909
and 119869(12059010
) = 12059013
This givesa contradiction since 120590
3
∘ 1205906
= 12059010
and 1205903
∘ 1205909
= 1205907
Neither ⟨120590
3
1205906
⟩ nor ⟨1205903
1205909
⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590
2
1205909
⟩ is not isomorphic to ⟨1205903
1205908
⟩since ⟨120590
2
1205909
⟩ has a unique pair of generators and ⟨1205903
1205908
⟩ =
⟨1205903
12059012
⟩Using A we getmdashisomorphic to previously discussed
onesmdashsemigroups ⟨1205904
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
⟨1205904
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
and ⟨1205904
1205907
⟩ =
⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
There exist minimal collec-tions of generators such as follows
⟨1205903
1205908
⟩ = ⟨1205903
1205906
1205909
⟩ = ⟨1205903
1205906
12059013
⟩
= ⟨1205903
1205909
12059010
⟩ = ⟨1205903
12059010
12059013
⟩
⟨1205904
1205907
⟩ = ⟨1205904
1205906
1205909
⟩ = ⟨1205904
1205909
12059010
⟩
= ⟨1205904
1205906
12059013
⟩ = ⟨1205904
12059010
12059013
⟩
(13)
73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590
2
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
and⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
= A[⟨1205902
1205903
⟩] andalso ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
and ⟨1205903
1205905
⟩ =
1205903
1205905
1205907
1205909
12059011
12059013
= A[⟨1205902
1205904
⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types
Journal of Mathematics 5
Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590
2
1205905
⟩ has 6 elements which are idempotentsIn ⟨1205903
1205904
⟩ there are seven semigroups which have threegenerators and have not two generators These are
(1) ⟨1205902
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩
(2) ⟨1205904
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩
(3) ⟨1205902
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩
(4) ⟨1205903
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩
(5) ⟨1205902
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩
(6) ⟨1205902
1205903
1205905
⟩
(7) ⟨1205902
1205904
1205905
⟩
8 Semigroups which Are Contained in ⟨12059031205904⟩
Groups ⟨1205902
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong ⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ have oneelement and are isomorphic
Groups ⟨1205906
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ monoids ⟨1205902
12059010
⟩ cong
⟨1205905
12059013
⟩ and also semigroups ⟨1205907
12059010
⟩ cong ⟨1205908
12059013
⟩ and⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ have two elements and Cayley tables asfollows
119860 119861
119860 119861 119860
119861 119860 119861
119860 119861
119860 119860 119861
119861 119861 119860
119860 119861
119860 119860 119861
119861 119860 119861
119860 119861
119860 119860 119860
119861 119861 119861
(14)
Monoids ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩ and also semigroups⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong ⟨1205904
⟩ havethree elements and Cayley tables as follows
119860 119861 119862
119860 119860 119861 119862
119861 119861 119862 119861
119862 119862 119861 119862
119860 119861 119862
119860 119860 119861 119862
119861 119862 119861 119862
119862 119862 119861 119862
119860 119861 119862
119860 119860 119862 119862
119861 119861 119861 119861
119862 119862 119862 119862
119860 119861 119862
119860 119861 119862 119861
119861 119862 119861 119862
119862 119861 119862 119861
(15)
Semigroups ⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and⟨1205907
1205908
⟩ have four elements and Cayley tables as follows
119860 119861 119862 119863
119860 119862 119863 119860 119861
119861 119860 119861 119862 119863
119862 119860 119861 119862 119863
119863 119862 119863 119860 119861
119860 119861 119862 119863
119860 119862 119860 119860 119862
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119861 119863 119863 119861
119860 119861 119862 119863
119860 119860 119862 119862 119860
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119863 119861 119861 119863
(16)
Semigroups ⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong
⟨1205904
1205906
⟩ have five elements Semigroups ⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590
0
1205902
1205905
⟩ has the followingCayley table
0 119860 119861 119862 119863 119864 119865
0 0 119860 119861 119862 119863 119864 119865
119860 119860 119860 119862 119862 119864 119864 119862
119861 119861 119863 119861 119865 119863 119863 119865
119862 119862 119864 119862 119862 119864 119864 119862
119863 119863 119863 119865 119865 119863 119863 119865
119864 119864 119864 119862 119862 119864 119864 119862
119865 119865 119863 119865 119865 119863 119863 119865
(17)
Preparing theCayley table for ⟨1205900
1205902
1205905
⟩weput1205900
= 0 1205902
=
119860 1205905
= 119861 1205907
= 119862 1205908
= 119863 12059010
= 119864 and 12059013
= 119865 This tableimmediately shows that semigroups ⟨120590
2
1205905
⟩ and ⟨1205900
1205902
1205905
⟩
have exactly two automorphisms These are identities andrestrictions of A
Semigroup ⟨1205906
1205909
⟩ has eight elements Semigroups⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ have nine ele-ments Semigroups ⟨120590
2
1205903
1205908
⟩ cong ⟨1205904
1205905
1205907
⟩ ⟨1205902
1205904
1205907
⟩ cong
⟨1205903
1205905
1205908
⟩ and ⟨1205902
1205905
1205906
⟩ have ten elements Semigroups⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩ have eleven elements In the endthe semigroup ⟨120590
3
1205904
⟩ has twelve elementsThus the semigroup ⟨120590
3
1205904
⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids
9 Viewing Semigroups Contained in M
91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +
6 Journal of Mathematics
4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590
3
1205904
⟩ fifty-seven monoidsformed by adding 120590
0
to a semigroup contained in ⟨1205903
1205904
⟩groups ⟨120590
0
⟩ ⟨1205901
⟩ and monoids ⟨1205901
1205906
⟩ = M1
and ⟨1205901
1205902
⟩ =
MThere are fifty-six types of nonisomorphic semigroups
in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590
3
1205904
⟩ twenty-six types of non-isomorphic monoids formed by adding 120590
0
to a semigroupcontained in ⟨120590
3
1205904
⟩ and alsoM1
andM Indeed adding 1205900
to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590
0
to a monoid contained in ⟨1205903
1205904
⟩ weget two new non-isomorphism types ofmonoids But adding1205900
to a group contained in ⟨1205903
1205904
⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590
2
12059010
⟩ or⟨1205902
1205906
⟩ The other two types areM1
andM
92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows
(1) ⟨1205902
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
(2) ⟨120590
2
1205903
1205905
⟩ = 1205902
1205903
1205905
1205906
12059013
(3) ⟨120590
2
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩ = 1205902
1205903
1205906
1205907
12059013
(4) ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
(5) ⟨120590
2
1205904
1205905
⟩ = 1205902
1205904
1205905
12059013
(6) ⟨120590
2
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩ = 1205902
1205904
1205906
1205907
12059013
(7) ⟨120590
2
1205905
⟩ = 1205902
1205905
1205907
1205908
12059010
12059013
(8) ⟨120590
2
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩ = 1205902
1205905
1205906
12059013
(9) ⟨120590
2
1205907
⟩ = 1205902
1205907
12059010
(10) ⟨120590
2
1205908
⟩ = 1205902
1205908
12059010
(11) ⟨120590
2
1205909
⟩ = ⟨1205902
1205906
12059013
⟩ = ⟨1205902
1205907
12059012
⟩ = ⟨1205902
1205908
12059011
⟩ = ⟨1205902
12059011
12059012
⟩ = ⟨1205902
12059011
12059013
⟩ = ⟨1205902
12059012
12059013
⟩ = 1205902
1205906
1205907
12059013
(12) ⟨120590
2
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(13) ⟨120590
2
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(14) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
(15) ⟨120590
3
⟩ = 1205903
1205907
12059011
(16) ⟨120590
3
1205904
⟩ = 1205902
1205903
12059013
(17) ⟨120590
3
1205905
⟩ = 1205903
1205905
1205907
1205909
12059011
12059013
(18) ⟨120590
3
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩ = 1205903
1205905
1205906
12059013
(19) ⟨120590
3
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(20) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(21) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
(22) ⟨120590
4
⟩ = 1205904
1205908
12059012
(23) ⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
(24) ⟨120590
4
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩ = 1205904
1205905
12059013
(25) ⟨120590
4
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
(26) ⟨120590
4
1205907
⟩ = ⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
(27) ⟨120590
4
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
(28) ⟨120590
5
1205906
⟩ = ⟨1205905
1205907
12059012
⟩ = ⟨1205905
1205908
12059011
⟩ = ⟨1205905
1205909
12059010
⟩ = ⟨1205905
12059010
12059011
⟩ = ⟨1205905
12059010
12059012
⟩ = ⟨1205905
12059011
12059012
⟩ = 1205905
1205906
12059013
(29) ⟨120590
5
1205907
⟩ = 1205905
1205907
12059013
(30) ⟨120590
5
1205908
⟩ = 1205905
1205908
12059013
(31) ⟨120590
5
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
(32) ⟨120590
5
12059011
⟩ = ⟨1205905
1205907
1205909
⟩ = 1205905
1205907
1205909
12059011
12059013
(33) ⟨120590
5
12059012
⟩ = ⟨1205905
1205908
1205909
⟩ = 1205905
1205908
1205909
12059012
12059013
(34) ⟨120590
6
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(35) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(36) ⟨120590
6
1205909
⟩ = ⟨1205906
12059013
⟩ = ⟨1205907
12059012
⟩ = ⟨1205908
12059011
⟩ = ⟨1205909
12059010
⟩ = ⟨12059011
12059012
⟩ = ⟨1205906
1205907
1205908
⟩ = ⟨1205907
1205908
1205909
⟩ =⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ = 1205906
1205907
12059013
(37) ⟨120590
7
1205908
⟩ = 1205907
1205908
12059010
12059013
(38) ⟨120590
7
1205909
⟩ = 1205907
1205909
12059011
12059013
(39) ⟨120590
7
12059010
⟩ = 1205907
12059010
(40) ⟨120590
7
12059013
⟩ = 1205907
12059013
(41) ⟨120590
8
1205909
⟩ = 1205908
1205909
12059012
12059013
(42) ⟨120590
8
12059010
⟩ = 1205908
12059010
(43) ⟨120590
8
12059013
⟩ = 1205908
12059013
93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on
(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups
(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590
0
added to 6 one-element groups and ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205908
12059010
⟩ cong ⟨1205907
12059013
⟩(iii) The monoid M contains 12 three-element monoids
with 5 isomorphism types These are 1205900
addedto 4 two-element groups and also ⟨120590
0
1205902
12059010
⟩ cong
⟨1205900
1205905
12059013
⟩ ⟨1205900
1205907
12059010
⟩ cong ⟨1205900
1205908
12059013
⟩⟨1205900
1205908
12059010
⟩ cong ⟨1205900
1205907
12059013
⟩ and ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩(iv) The monoidM contains 6 three-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong
⟨1205904
⟩
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Mathematics
⟨1205909
12059010
⟩ or ⟨12059011
12059012
⟩ Also ⟨1205906
1205909
⟩ = ⟨1205906
1205907
1205908
⟩ =
⟨1205907
1205908
1205909
⟩ = ⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ These rep-resentations exhaust all minimal collections of the Kura-towski operations which generate ⟨120590
6
1205909
⟩ Other semigroupsincluded in ⟨120590
6
1205909
⟩ have one or two minimal collectionof generators One generator has groups ⟨120590
6
⟩ = 1205906
12059010
⟨1205909
⟩ = 1205909
12059013
⟨12059011
⟩ = 1205907
12059011
and ⟨12059012
⟩ = 1205908
12059012
Each of them has exactly two elements so they are isomor-phic Semigroups ⟨120590
7
12059010
⟩ ⟨1205907
12059013
⟩ ⟨1205908
12059010
⟩ ⟨1205908
12059013
⟩ and⟨1205907
1205908
⟩ are discussed in the previous section Contained in⟨1205906
1205909
⟩ and not previously discussed semigroups are ⟨1205906
1205907
⟩⟨1205906
1205908
⟩⟨1205907
1205909
⟩ and ⟨1205908
1205909
⟩ We leave the reader to verifythat the following are all possible pairs of Kuratowski oper-ations which constitute a minimal collection of generatorsfor semigroups contained in ⟨120590
6
1205909
⟩ but different from thewhole One has
(i) ⟨1205906
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(ii) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(iii) ⟨120590
7
1205909
⟩ = ⟨1205909
12059011
⟩ = ⟨12059011
12059013
⟩ = 1205907
1205909
12059011
12059013
(iv) ⟨120590
8
1205909
⟩ = ⟨1205909
12059012
⟩ = ⟨12059012
12059013
⟩ = 1205908
1205909
12059012
12059013
Proposition 4 Semigroups ⟨1205906
1205907
⟩ and ⟨1205908
1205909
⟩ are isomor-phic and also semigroups ⟨120590
6
1205908
⟩ and ⟨1205907
1205909
⟩ are isomorphicbut semigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ are not isomorphic
Proof Isomorphisms are defined by A Suppose 119869 ⟨1205906
1205907
⟩ rarr ⟨1205906
1205908
⟩ is an isomorphism Thus 119869[⟨1205907
12059010
⟩] =
⟨1205908
12059010
⟩ Given 119869(1205907
) = 1205908
we get 12059010
= 119869(12059010
) = 119869(1205907
∘
12059010
) = 1205908
∘ 12059010
= 1205908
But 119869(1205907
) = 12059010
implies 12059010
= 119869(1205907
) =
119869(12059010
∘ 1205907
) = 1205908
∘ 12059010
= 1205908
Both possibilities lead to acontradiction
So ⟨1205906
1205909
⟩ contains eighteen different semigroups witheight non-isomorphism types Indeed these are four iso-morphic groups with exactly one element ⟨120590
7
⟩ cong ⟨1205908
⟩ cong
⟨12059010
⟩ cong ⟨12059013
⟩ four isomorphic groups with exactly twoelements ⟨120590
6
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ two pairs ofisomorphic semigroups with exactly two elements ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ and five semigroupswith exactly four elements ⟨120590
6
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong
⟨1205907
1205909
⟩ ⟨1205907
1205908
⟩ and also ⟨1205906
1205909
⟩
7 Remaining Semigroups in ⟨1205903 1205904⟩
We have yet to discuss semigroups included in ⟨1205903
1205904
⟩ notincluded in ⟨120590
2
1205905
⟩ and containing at least one of Kuratowskioperation 120590
2
1205903
1205904
or 1205905
It will be discussed up to theisomorphism A Obviously ⟨120590
2
⟩ = 1205902
and ⟨1205905
⟩ = 1205905
aregroups
71 Extensions of ⟨1205902⟩ and ⟨1205905⟩ with Elements of ⟨12059061205909⟩Monoids ⟨120590
2
1205906
⟩ = 1205902
1205906
12059010
and ⟨1205902
12059010
⟩ = 1205902
12059010
havedifferent numbers of elements Also ⟨120590
2
1205906
⟩ is isomorphic to⟨1205900
1205906
⟩ Nonisomorphic semigroups ⟨1205902
1205907
⟩ and ⟨1205902
1205908
⟩ arediscussed above The following three semigroups
(i) ⟨1205902
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(ii) ⟨1205902
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(iii) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
are not monoids They are not isomorphic Indeed anyisomorphism between these semigroups would lead an iso-morphism between ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ or ⟨1205907
1205908
⟩ This isimpossible by Proposition 4 and because ⟨120590
7
1205908
⟩ consists ofidempotents but 120590
6
is not an idempotent The nine-elementsemigroup on the set 120590
2
1205906
1205907
12059013
is represented as⟨1205902
1205909
⟩ So we have added four new semigroups which arenot isomorphic with the semigroups previously discussedThese are ⟨120590
2
12059011
⟩ ⟨1205902
12059012
⟩ ⟨1205902
12059013
⟩ and ⟨1205902
1205909
⟩Using A we have described eight semigroupsmdasheach
one isomorphic to a semigroup previously discussedmdashwhichcontains 120590
5
and elements (at least one) of ⟨1205906
1205909
⟩ Collec-tions of generators ⟨120590
2
1205906
12059013
⟩ ⟨1205902
1205907
12059012
⟩ ⟨1205902
1205908
12059011
⟩⟨1205902
12059011
12059012
⟩ ⟨1205902
12059011
12059013
⟩ and ⟨1205902
12059012
12059013
⟩ are minimalin ⟨120590
2
1205909
⟩ Also collections of generators ⟨1205905
1205907
12059012
⟩⟨1205905
1205908
12059011
⟩ ⟨1205905
1205909
12059010
⟩ ⟨1205905
12059010
12059011
⟩ ⟨1205905
12059010
12059012
⟩ and⟨1205905
12059011
12059012
⟩ are minimal in ⟨1205905
1205906
⟩
72 Extensions of ⟨1205903⟩ and ⟨1205904⟩ by Elements from the Semi-group ⟨12059061205909⟩ Semigroups ⟨120590
3
⟩ = 1205903
1205907
12059011
and ⟨1205904
⟩ =
1205904
1205908
12059012
are isomorphic by A They are not monoids Thesemigroup ⟨120590
3
⟩ can be extended using elements of ⟨1205906
1205909
⟩in three following ways
(i) ⟨1205903
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(ii) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(iii) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
Semigroups ⟨1205903
1205906
⟩ and ⟨1205903
1205909
⟩ are not isomorphic Indeedsuppose 119869 ⟨120590
3
1205906
⟩ rarr ⟨1205903
1205909
⟩ is an isomorphismThus 119869 isthe identity on ⟨120590
3
⟩ 119869(1205906
) = 1205909
and 119869(12059010
) = 12059013
This givesa contradiction since 120590
3
∘ 1205906
= 12059010
and 1205903
∘ 1205909
= 1205907
Neither ⟨120590
3
1205906
⟩ nor ⟨1205903
1205909
⟩ has a minimal collection ofgenerators with three elements so they give new isomor-phism types Also ⟨120590
2
1205909
⟩ is not isomorphic to ⟨1205903
1205908
⟩since ⟨120590
2
1205909
⟩ has a unique pair of generators and ⟨1205903
1205908
⟩ =
⟨1205903
12059012
⟩Using A we getmdashisomorphic to previously discussed
onesmdashsemigroups ⟨1205904
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
⟨1205904
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
and ⟨1205904
1205907
⟩ =
⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
There exist minimal collec-tions of generators such as follows
⟨1205903
1205908
⟩ = ⟨1205903
1205906
1205909
⟩ = ⟨1205903
1205906
12059013
⟩
= ⟨1205903
1205909
12059010
⟩ = ⟨1205903
12059010
12059013
⟩
⟨1205904
1205907
⟩ = ⟨1205904
1205906
1205909
⟩ = ⟨1205904
1205909
12059010
⟩
= ⟨1205904
1205906
12059013
⟩ = ⟨1205904
12059010
12059013
⟩
(13)
73 More Generators from the Set 1205902120590312059041205905 Nowwe check that ⟨120590
2
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
and⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
= A[⟨1205902
1205903
⟩] andalso ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
and ⟨1205903
1205905
⟩ =
1205903
1205905
1205907
1205909
12059011
12059013
= A[⟨1205902
1205904
⟩] are two pairs of iso-morphic semigroupswhich give two new isomorphism types
Journal of Mathematics 5
Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590
2
1205905
⟩ has 6 elements which are idempotentsIn ⟨1205903
1205904
⟩ there are seven semigroups which have threegenerators and have not two generators These are
(1) ⟨1205902
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩
(2) ⟨1205904
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩
(3) ⟨1205902
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩
(4) ⟨1205903
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩
(5) ⟨1205902
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩
(6) ⟨1205902
1205903
1205905
⟩
(7) ⟨1205902
1205904
1205905
⟩
8 Semigroups which Are Contained in ⟨12059031205904⟩
Groups ⟨1205902
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong ⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ have oneelement and are isomorphic
Groups ⟨1205906
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ monoids ⟨1205902
12059010
⟩ cong
⟨1205905
12059013
⟩ and also semigroups ⟨1205907
12059010
⟩ cong ⟨1205908
12059013
⟩ and⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ have two elements and Cayley tables asfollows
119860 119861
119860 119861 119860
119861 119860 119861
119860 119861
119860 119860 119861
119861 119861 119860
119860 119861
119860 119860 119861
119861 119860 119861
119860 119861
119860 119860 119860
119861 119861 119861
(14)
Monoids ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩ and also semigroups⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong ⟨1205904
⟩ havethree elements and Cayley tables as follows
119860 119861 119862
119860 119860 119861 119862
119861 119861 119862 119861
119862 119862 119861 119862
119860 119861 119862
119860 119860 119861 119862
119861 119862 119861 119862
119862 119862 119861 119862
119860 119861 119862
119860 119860 119862 119862
119861 119861 119861 119861
119862 119862 119862 119862
119860 119861 119862
119860 119861 119862 119861
119861 119862 119861 119862
119862 119861 119862 119861
(15)
Semigroups ⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and⟨1205907
1205908
⟩ have four elements and Cayley tables as follows
119860 119861 119862 119863
119860 119862 119863 119860 119861
119861 119860 119861 119862 119863
119862 119860 119861 119862 119863
119863 119862 119863 119860 119861
119860 119861 119862 119863
119860 119862 119860 119860 119862
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119861 119863 119863 119861
119860 119861 119862 119863
119860 119860 119862 119862 119860
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119863 119861 119861 119863
(16)
Semigroups ⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong
⟨1205904
1205906
⟩ have five elements Semigroups ⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590
0
1205902
1205905
⟩ has the followingCayley table
0 119860 119861 119862 119863 119864 119865
0 0 119860 119861 119862 119863 119864 119865
119860 119860 119860 119862 119862 119864 119864 119862
119861 119861 119863 119861 119865 119863 119863 119865
119862 119862 119864 119862 119862 119864 119864 119862
119863 119863 119863 119865 119865 119863 119863 119865
119864 119864 119864 119862 119862 119864 119864 119862
119865 119865 119863 119865 119865 119863 119863 119865
(17)
Preparing theCayley table for ⟨1205900
1205902
1205905
⟩weput1205900
= 0 1205902
=
119860 1205905
= 119861 1205907
= 119862 1205908
= 119863 12059010
= 119864 and 12059013
= 119865 This tableimmediately shows that semigroups ⟨120590
2
1205905
⟩ and ⟨1205900
1205902
1205905
⟩
have exactly two automorphisms These are identities andrestrictions of A
Semigroup ⟨1205906
1205909
⟩ has eight elements Semigroups⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ have nine ele-ments Semigroups ⟨120590
2
1205903
1205908
⟩ cong ⟨1205904
1205905
1205907
⟩ ⟨1205902
1205904
1205907
⟩ cong
⟨1205903
1205905
1205908
⟩ and ⟨1205902
1205905
1205906
⟩ have ten elements Semigroups⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩ have eleven elements In the endthe semigroup ⟨120590
3
1205904
⟩ has twelve elementsThus the semigroup ⟨120590
3
1205904
⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids
9 Viewing Semigroups Contained in M
91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +
6 Journal of Mathematics
4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590
3
1205904
⟩ fifty-seven monoidsformed by adding 120590
0
to a semigroup contained in ⟨1205903
1205904
⟩groups ⟨120590
0
⟩ ⟨1205901
⟩ and monoids ⟨1205901
1205906
⟩ = M1
and ⟨1205901
1205902
⟩ =
MThere are fifty-six types of nonisomorphic semigroups
in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590
3
1205904
⟩ twenty-six types of non-isomorphic monoids formed by adding 120590
0
to a semigroupcontained in ⟨120590
3
1205904
⟩ and alsoM1
andM Indeed adding 1205900
to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590
0
to a monoid contained in ⟨1205903
1205904
⟩ weget two new non-isomorphism types ofmonoids But adding1205900
to a group contained in ⟨1205903
1205904
⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590
2
12059010
⟩ or⟨1205902
1205906
⟩ The other two types areM1
andM
92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows
(1) ⟨1205902
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
(2) ⟨120590
2
1205903
1205905
⟩ = 1205902
1205903
1205905
1205906
12059013
(3) ⟨120590
2
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩ = 1205902
1205903
1205906
1205907
12059013
(4) ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
(5) ⟨120590
2
1205904
1205905
⟩ = 1205902
1205904
1205905
12059013
(6) ⟨120590
2
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩ = 1205902
1205904
1205906
1205907
12059013
(7) ⟨120590
2
1205905
⟩ = 1205902
1205905
1205907
1205908
12059010
12059013
(8) ⟨120590
2
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩ = 1205902
1205905
1205906
12059013
(9) ⟨120590
2
1205907
⟩ = 1205902
1205907
12059010
(10) ⟨120590
2
1205908
⟩ = 1205902
1205908
12059010
(11) ⟨120590
2
1205909
⟩ = ⟨1205902
1205906
12059013
⟩ = ⟨1205902
1205907
12059012
⟩ = ⟨1205902
1205908
12059011
⟩ = ⟨1205902
12059011
12059012
⟩ = ⟨1205902
12059011
12059013
⟩ = ⟨1205902
12059012
12059013
⟩ = 1205902
1205906
1205907
12059013
(12) ⟨120590
2
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(13) ⟨120590
2
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(14) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
(15) ⟨120590
3
⟩ = 1205903
1205907
12059011
(16) ⟨120590
3
1205904
⟩ = 1205902
1205903
12059013
(17) ⟨120590
3
1205905
⟩ = 1205903
1205905
1205907
1205909
12059011
12059013
(18) ⟨120590
3
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩ = 1205903
1205905
1205906
12059013
(19) ⟨120590
3
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(20) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(21) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
(22) ⟨120590
4
⟩ = 1205904
1205908
12059012
(23) ⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
(24) ⟨120590
4
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩ = 1205904
1205905
12059013
(25) ⟨120590
4
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
(26) ⟨120590
4
1205907
⟩ = ⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
(27) ⟨120590
4
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
(28) ⟨120590
5
1205906
⟩ = ⟨1205905
1205907
12059012
⟩ = ⟨1205905
1205908
12059011
⟩ = ⟨1205905
1205909
12059010
⟩ = ⟨1205905
12059010
12059011
⟩ = ⟨1205905
12059010
12059012
⟩ = ⟨1205905
12059011
12059012
⟩ = 1205905
1205906
12059013
(29) ⟨120590
5
1205907
⟩ = 1205905
1205907
12059013
(30) ⟨120590
5
1205908
⟩ = 1205905
1205908
12059013
(31) ⟨120590
5
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
(32) ⟨120590
5
12059011
⟩ = ⟨1205905
1205907
1205909
⟩ = 1205905
1205907
1205909
12059011
12059013
(33) ⟨120590
5
12059012
⟩ = ⟨1205905
1205908
1205909
⟩ = 1205905
1205908
1205909
12059012
12059013
(34) ⟨120590
6
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(35) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(36) ⟨120590
6
1205909
⟩ = ⟨1205906
12059013
⟩ = ⟨1205907
12059012
⟩ = ⟨1205908
12059011
⟩ = ⟨1205909
12059010
⟩ = ⟨12059011
12059012
⟩ = ⟨1205906
1205907
1205908
⟩ = ⟨1205907
1205908
1205909
⟩ =⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ = 1205906
1205907
12059013
(37) ⟨120590
7
1205908
⟩ = 1205907
1205908
12059010
12059013
(38) ⟨120590
7
1205909
⟩ = 1205907
1205909
12059011
12059013
(39) ⟨120590
7
12059010
⟩ = 1205907
12059010
(40) ⟨120590
7
12059013
⟩ = 1205907
12059013
(41) ⟨120590
8
1205909
⟩ = 1205908
1205909
12059012
12059013
(42) ⟨120590
8
12059010
⟩ = 1205908
12059010
(43) ⟨120590
8
12059013
⟩ = 1205908
12059013
93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on
(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups
(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590
0
added to 6 one-element groups and ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205908
12059010
⟩ cong ⟨1205907
12059013
⟩(iii) The monoid M contains 12 three-element monoids
with 5 isomorphism types These are 1205900
addedto 4 two-element groups and also ⟨120590
0
1205902
12059010
⟩ cong
⟨1205900
1205905
12059013
⟩ ⟨1205900
1205907
12059010
⟩ cong ⟨1205900
1205908
12059013
⟩⟨1205900
1205908
12059010
⟩ cong ⟨1205900
1205907
12059013
⟩ and ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩(iv) The monoidM contains 6 three-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong
⟨1205904
⟩
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Journal of Mathematics 5
Each of these semigroups has six element so in M thereare five six-element semigroups of three isomorphism typessince ⟨120590
2
1205905
⟩ has 6 elements which are idempotentsIn ⟨1205903
1205904
⟩ there are seven semigroups which have threegenerators and have not two generators These are
(1) ⟨1205902
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩
(2) ⟨1205904
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩
(3) ⟨1205902
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩
(4) ⟨1205903
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩
(5) ⟨1205902
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩
(6) ⟨1205902
1205903
1205905
⟩
(7) ⟨1205902
1205904
1205905
⟩
8 Semigroups which Are Contained in ⟨12059031205904⟩
Groups ⟨1205902
⟩ cong ⟨1205905
⟩ cong ⟨1205907
⟩ cong ⟨1205908
⟩ cong ⟨12059010
⟩ cong ⟨12059013
⟩ have oneelement and are isomorphic
Groups ⟨1205906
⟩ cong ⟨1205909
⟩ cong ⟨12059011
⟩ cong ⟨12059012
⟩ monoids ⟨1205902
12059010
⟩ cong
⟨1205905
12059013
⟩ and also semigroups ⟨1205907
12059010
⟩ cong ⟨1205908
12059013
⟩ and⟨1205907
12059013
⟩ cong ⟨1205908
12059010
⟩ have two elements and Cayley tables asfollows
119860 119861
119860 119861 119860
119861 119860 119861
119860 119861
119860 119860 119861
119861 119861 119860
119860 119861
119860 119860 119861
119861 119860 119861
119860 119861
119860 119860 119860
119861 119861 119861
(14)
Monoids ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩ and also semigroups⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong ⟨1205904
⟩ havethree elements and Cayley tables as follows
119860 119861 119862
119860 119860 119861 119862
119861 119861 119862 119861
119862 119862 119861 119862
119860 119861 119862
119860 119860 119861 119862
119861 119862 119861 119862
119862 119862 119861 119862
119860 119861 119862
119860 119860 119862 119862
119861 119861 119861 119861
119862 119862 119862 119862
119860 119861 119862
119860 119861 119862 119861
119861 119862 119861 119862
119862 119861 119862 119861
(15)
Semigroups ⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and⟨1205907
1205908
⟩ have four elements and Cayley tables as follows
119860 119861 119862 119863
119860 119862 119863 119860 119861
119861 119860 119861 119862 119863
119862 119860 119861 119862 119863
119863 119862 119863 119860 119861
119860 119861 119862 119863
119860 119862 119860 119860 119862
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119861 119863 119863 119861
119860 119861 119862 119863
119860 119860 119862 119862 119860
119861 119863 119861 119861 119863
119862 119860 119862 119862 119860
119863 119863 119861 119861 119863
(16)
Semigroups ⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩⟨1205902
12059013
⟩ cong ⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong
⟨1205904
1205906
⟩ have five elements Semigroups ⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩ have six elements Construc-tion of Cayley tables for these semigroups as well as othersemigroups we leave it to the readers One can prepare suchtables removing from the Cayley table for M some columnsand rows Then change Kuratowski operations onto lettersof the alphabet For example ⟨120590
0
1205902
1205905
⟩ has the followingCayley table
0 119860 119861 119862 119863 119864 119865
0 0 119860 119861 119862 119863 119864 119865
119860 119860 119860 119862 119862 119864 119864 119862
119861 119861 119863 119861 119865 119863 119863 119865
119862 119862 119864 119862 119862 119864 119864 119862
119863 119863 119863 119865 119865 119863 119863 119865
119864 119864 119864 119862 119862 119864 119864 119862
119865 119865 119863 119865 119865 119863 119863 119865
(17)
Preparing theCayley table for ⟨1205900
1205902
1205905
⟩weput1205900
= 0 1205902
=
119860 1205905
= 119861 1205907
= 119862 1205908
= 119863 12059010
= 119864 and 12059013
= 119865 This tableimmediately shows that semigroups ⟨120590
2
1205905
⟩ and ⟨1205900
1205902
1205905
⟩
have exactly two automorphisms These are identities andrestrictions of A
Semigroup ⟨1205906
1205909
⟩ has eight elements Semigroups⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ have nine ele-ments Semigroups ⟨120590
2
1205903
1205908
⟩ cong ⟨1205904
1205905
1205907
⟩ ⟨1205902
1205904
1205907
⟩ cong
⟨1205903
1205905
1205908
⟩ and ⟨1205902
1205905
1205906
⟩ have ten elements Semigroups⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩ have eleven elements In the endthe semigroup ⟨120590
3
1205904
⟩ has twelve elementsThus the semigroup ⟨120590
3
1205904
⟩ includes fifty-seven semi-groups amongwhich there are ten groups fourteenmonoidsand also forty-three semigroups which are not monoidsThese semigroups consist of twenty-eight types of noniso-morphic semigroups two non-isomorphism types of groupstwo non-isomorphism types of monoids which are notgroups and twenty four non-isomorphism types of semi-groups which are not monoids
9 Viewing Semigroups Contained in M
91 Descriptive Data on Semigroups which Are Contained inM There are one hundred eighteen that is 118 = 2 sdot 57 +
6 Journal of Mathematics
4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590
3
1205904
⟩ fifty-seven monoidsformed by adding 120590
0
to a semigroup contained in ⟨1205903
1205904
⟩groups ⟨120590
0
⟩ ⟨1205901
⟩ and monoids ⟨1205901
1205906
⟩ = M1
and ⟨1205901
1205902
⟩ =
MThere are fifty-six types of nonisomorphic semigroups
in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590
3
1205904
⟩ twenty-six types of non-isomorphic monoids formed by adding 120590
0
to a semigroupcontained in ⟨120590
3
1205904
⟩ and alsoM1
andM Indeed adding 1205900
to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590
0
to a monoid contained in ⟨1205903
1205904
⟩ weget two new non-isomorphism types ofmonoids But adding1205900
to a group contained in ⟨1205903
1205904
⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590
2
12059010
⟩ or⟨1205902
1205906
⟩ The other two types areM1
andM
92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows
(1) ⟨1205902
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
(2) ⟨120590
2
1205903
1205905
⟩ = 1205902
1205903
1205905
1205906
12059013
(3) ⟨120590
2
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩ = 1205902
1205903
1205906
1205907
12059013
(4) ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
(5) ⟨120590
2
1205904
1205905
⟩ = 1205902
1205904
1205905
12059013
(6) ⟨120590
2
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩ = 1205902
1205904
1205906
1205907
12059013
(7) ⟨120590
2
1205905
⟩ = 1205902
1205905
1205907
1205908
12059010
12059013
(8) ⟨120590
2
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩ = 1205902
1205905
1205906
12059013
(9) ⟨120590
2
1205907
⟩ = 1205902
1205907
12059010
(10) ⟨120590
2
1205908
⟩ = 1205902
1205908
12059010
(11) ⟨120590
2
1205909
⟩ = ⟨1205902
1205906
12059013
⟩ = ⟨1205902
1205907
12059012
⟩ = ⟨1205902
1205908
12059011
⟩ = ⟨1205902
12059011
12059012
⟩ = ⟨1205902
12059011
12059013
⟩ = ⟨1205902
12059012
12059013
⟩ = 1205902
1205906
1205907
12059013
(12) ⟨120590
2
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(13) ⟨120590
2
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(14) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
(15) ⟨120590
3
⟩ = 1205903
1205907
12059011
(16) ⟨120590
3
1205904
⟩ = 1205902
1205903
12059013
(17) ⟨120590
3
1205905
⟩ = 1205903
1205905
1205907
1205909
12059011
12059013
(18) ⟨120590
3
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩ = 1205903
1205905
1205906
12059013
(19) ⟨120590
3
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(20) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(21) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
(22) ⟨120590
4
⟩ = 1205904
1205908
12059012
(23) ⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
(24) ⟨120590
4
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩ = 1205904
1205905
12059013
(25) ⟨120590
4
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
(26) ⟨120590
4
1205907
⟩ = ⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
(27) ⟨120590
4
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
(28) ⟨120590
5
1205906
⟩ = ⟨1205905
1205907
12059012
⟩ = ⟨1205905
1205908
12059011
⟩ = ⟨1205905
1205909
12059010
⟩ = ⟨1205905
12059010
12059011
⟩ = ⟨1205905
12059010
12059012
⟩ = ⟨1205905
12059011
12059012
⟩ = 1205905
1205906
12059013
(29) ⟨120590
5
1205907
⟩ = 1205905
1205907
12059013
(30) ⟨120590
5
1205908
⟩ = 1205905
1205908
12059013
(31) ⟨120590
5
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
(32) ⟨120590
5
12059011
⟩ = ⟨1205905
1205907
1205909
⟩ = 1205905
1205907
1205909
12059011
12059013
(33) ⟨120590
5
12059012
⟩ = ⟨1205905
1205908
1205909
⟩ = 1205905
1205908
1205909
12059012
12059013
(34) ⟨120590
6
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(35) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(36) ⟨120590
6
1205909
⟩ = ⟨1205906
12059013
⟩ = ⟨1205907
12059012
⟩ = ⟨1205908
12059011
⟩ = ⟨1205909
12059010
⟩ = ⟨12059011
12059012
⟩ = ⟨1205906
1205907
1205908
⟩ = ⟨1205907
1205908
1205909
⟩ =⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ = 1205906
1205907
12059013
(37) ⟨120590
7
1205908
⟩ = 1205907
1205908
12059010
12059013
(38) ⟨120590
7
1205909
⟩ = 1205907
1205909
12059011
12059013
(39) ⟨120590
7
12059010
⟩ = 1205907
12059010
(40) ⟨120590
7
12059013
⟩ = 1205907
12059013
(41) ⟨120590
8
1205909
⟩ = 1205908
1205909
12059012
12059013
(42) ⟨120590
8
12059010
⟩ = 1205908
12059010
(43) ⟨120590
8
12059013
⟩ = 1205908
12059013
93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on
(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups
(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590
0
added to 6 one-element groups and ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205908
12059010
⟩ cong ⟨1205907
12059013
⟩(iii) The monoid M contains 12 three-element monoids
with 5 isomorphism types These are 1205900
addedto 4 two-element groups and also ⟨120590
0
1205902
12059010
⟩ cong
⟨1205900
1205905
12059013
⟩ ⟨1205900
1205907
12059010
⟩ cong ⟨1205900
1205908
12059013
⟩⟨1205900
1205908
12059010
⟩ cong ⟨1205900
1205907
12059013
⟩ and ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩(iv) The monoidM contains 6 three-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong
⟨1205904
⟩
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
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6 Journal of Mathematics
4 semigroups which are contained in M These are fifty-seven semigroups contained in ⟨120590
3
1205904
⟩ fifty-seven monoidsformed by adding 120590
0
to a semigroup contained in ⟨1205903
1205904
⟩groups ⟨120590
0
⟩ ⟨1205901
⟩ and monoids ⟨1205901
1205906
⟩ = M1
and ⟨1205901
1205902
⟩ =
MThere are fifty-six types of nonisomorphic semigroups
in M These are twenty-eight non-isomorphism types ofsemigroups contained in ⟨120590
3
1205904
⟩ twenty-six types of non-isomorphic monoids formed by adding 120590
0
to a semigroupcontained in ⟨120590
3
1205904
⟩ and alsoM1
andM Indeed adding 1205900
to a semigroup which is not a monoid we obtain a monoidIn this way we get twenty-four types of nonisomorphicmonoids Adding 120590
0
to a monoid contained in ⟨1205903
1205904
⟩ weget two new non-isomorphism types ofmonoids But adding1205900
to a group contained in ⟨1205903
1205904
⟩ we get no new type ofmonoid since we get a monoid isomorphic to ⟨120590
2
12059010
⟩ or⟨1205902
1205906
⟩ The other two types areM1
andM
92 Semigroups which Are Not Monoids Below we havereproduced using the smallest number of generators andthe dictionary order a list of all 43 semigroups which areincluded in theM as follows
(1) ⟨1205902
1205903
⟩ = 1205902
1205903
1205906
1205907
12059010
12059011
(2) ⟨120590
2
1205903
1205905
⟩ = 1205902
1205903
1205905
1205906
12059013
(3) ⟨120590
2
1205903
1205908
⟩ = ⟨1205902
1205903
1205909
⟩ = ⟨1205902
1205903
12059012
⟩ = ⟨1205902
1205903
12059013
⟩ = 1205902
1205903
1205906
1205907
12059013
(4) ⟨120590
2
1205904
⟩ = 1205902
1205904
1205906
1205908
12059010
12059012
(5) ⟨120590
2
1205904
1205905
⟩ = 1205902
1205904
1205905
12059013
(6) ⟨120590
2
1205904
1205907
⟩ = ⟨1205902
1205904
1205909
⟩ = ⟨1205902
1205904
12059011
⟩ = ⟨1205902
1205904
12059013
⟩ = 1205902
1205904
1205906
1205907
12059013
(7) ⟨120590
2
1205905
⟩ = 1205902
1205905
1205907
1205908
12059010
12059013
(8) ⟨120590
2
1205905
1205906
⟩ = ⟨1205902
1205905
1205909
⟩ = ⟨1205902
1205905
12059011
⟩ = ⟨1205902
1205905
12059012
⟩ = 1205902
1205905
1205906
12059013
(9) ⟨120590
2
1205907
⟩ = 1205902
1205907
12059010
(10) ⟨120590
2
1205908
⟩ = 1205902
1205908
12059010
(11) ⟨120590
2
1205909
⟩ = ⟨1205902
1205906
12059013
⟩ = ⟨1205902
1205907
12059012
⟩ = ⟨1205902
1205908
12059011
⟩ = ⟨1205902
12059011
12059012
⟩ = ⟨1205902
12059011
12059013
⟩ = ⟨1205902
12059012
12059013
⟩ = 1205902
1205906
1205907
12059013
(12) ⟨120590
2
12059011
⟩ = ⟨1205902
1205906
1205907
⟩ = 1205902
1205906
1205907
12059010
12059011
(13) ⟨120590
2
12059012
⟩ = ⟨1205902
1205906
1205908
⟩ = 1205902
1205906
1205908
12059010
12059012
(14) ⟨120590
2
12059013
⟩ = ⟨1205902
1205907
1205908
⟩ = 1205902
1205907
1205908
12059010
12059013
(15) ⟨120590
3
⟩ = 1205903
1205907
12059011
(16) ⟨120590
3
1205904
⟩ = 1205902
1205903
12059013
(17) ⟨120590
3
1205905
⟩ = 1205903
1205905
1205907
1205909
12059011
12059013
(18) ⟨120590
3
1205905
1205906
⟩ = ⟨1205903
1205905
1205908
⟩ = ⟨1205903
1205905
12059010
⟩ = ⟨1205903
1205905
12059012
⟩ = 1205903
1205905
1205906
12059013
(19) ⟨120590
3
1205906
⟩ = ⟨1205903
12059010
⟩ = 1205903
1205906
1205907
12059010
12059011
(20) ⟨120590
3
1205908
⟩ = ⟨1205903
12059012
⟩ = 1205903
1205906
1205907
12059013
(21) ⟨120590
3
1205909
⟩ = ⟨1205903
12059013
⟩ = 1205903
1205907
1205909
12059011
12059013
(22) ⟨120590
4
⟩ = 1205904
1205908
12059012
(23) ⟨1205904
1205905
⟩ = 1205904
1205905
1205908
1205909
12059012
12059013
(24) ⟨120590
4
1205905
1205906
⟩ = ⟨1205904
1205905
1205907
⟩ = ⟨1205904
1205905
12059010
⟩ = ⟨1205904
1205905
12059011
⟩ = 1205904
1205905
12059013
(25) ⟨120590
4
1205906
⟩ = ⟨1205904
12059010
⟩ = 1205904
1205906
1205908
12059010
12059012
(26) ⟨120590
4
1205907
⟩ = ⟨1205904
12059011
⟩ = 1205904
1205906
1205907
12059013
(27) ⟨120590
4
1205909
⟩ = ⟨1205904
12059013
⟩ = 1205904
1205908
1205909
12059012
12059013
(28) ⟨120590
5
1205906
⟩ = ⟨1205905
1205907
12059012
⟩ = ⟨1205905
1205908
12059011
⟩ = ⟨1205905
1205909
12059010
⟩ = ⟨1205905
12059010
12059011
⟩ = ⟨1205905
12059010
12059012
⟩ = ⟨1205905
12059011
12059012
⟩ = 1205905
1205906
12059013
(29) ⟨120590
5
1205907
⟩ = 1205905
1205907
12059013
(30) ⟨120590
5
1205908
⟩ = 1205905
1205908
12059013
(31) ⟨120590
5
12059010
⟩ = ⟨1205905
1205907
1205908
⟩ = 1205905
1205907
1205908
12059010
12059013
(32) ⟨120590
5
12059011
⟩ = ⟨1205905
1205907
1205909
⟩ = 1205905
1205907
1205909
12059011
12059013
(33) ⟨120590
5
12059012
⟩ = ⟨1205905
1205908
1205909
⟩ = 1205905
1205908
1205909
12059012
12059013
(34) ⟨120590
6
1205907
⟩ = ⟨1205906
12059011
⟩ = ⟨12059010
12059011
⟩ = 1205906
1205907
12059010
12059011
(35) ⟨120590
6
1205908
⟩ = ⟨1205906
12059012
⟩ = ⟨12059010
12059012
⟩ = 1205906
1205908
12059010
12059012
(36) ⟨120590
6
1205909
⟩ = ⟨1205906
12059013
⟩ = ⟨1205907
12059012
⟩ = ⟨1205908
12059011
⟩ = ⟨1205909
12059010
⟩ = ⟨12059011
12059012
⟩ = ⟨1205906
1205907
1205908
⟩ = ⟨1205907
1205908
1205909
⟩ =⟨12059010
12059011
12059013
⟩ = ⟨12059010
12059012
12059013
⟩ = 1205906
1205907
12059013
(37) ⟨120590
7
1205908
⟩ = 1205907
1205908
12059010
12059013
(38) ⟨120590
7
1205909
⟩ = 1205907
1205909
12059011
12059013
(39) ⟨120590
7
12059010
⟩ = 1205907
12059010
(40) ⟨120590
7
12059013
⟩ = 1205907
12059013
(41) ⟨120590
8
1205909
⟩ = 1205908
1205909
12059012
12059013
(42) ⟨120590
8
12059010
⟩ = 1205908
12059010
(43) ⟨120590
8
12059013
⟩ = 1205908
12059013
93 Isomorphism Types of Semigroups Contained in M Sys-tematize the list of all isomorphism types of semigroupscontained in themonoidM Isomorphisms which are restric-tions of the isomorphism A will be regarded as self-evidentand therefore they will not be commented on
(i) The monoid M contains 12 groups with 2 isomor-phism types These are 7 one-element groups and 5two-element groups
(ii) ThemonoidM contains 8 two-element monoids with2 isomorphism types These are 120590
0
added to 6 one-element groups and ⟨120590
2
12059010
⟩ cong ⟨1205905
12059013
⟩ Also itcontains 4 two-element semigroups not monoidswith 2 isomorphism types These are ⟨120590
7
12059010
⟩ cong
⟨1205908
12059013
⟩ and ⟨1205908
12059010
⟩ cong ⟨1205907
12059013
⟩(iii) The monoid M contains 12 three-element monoids
with 5 isomorphism types These are 1205900
addedto 4 two-element groups and also ⟨120590
0
1205902
12059010
⟩ cong
⟨1205900
1205905
12059013
⟩ ⟨1205900
1205907
12059010
⟩ cong ⟨1205900
1205908
12059013
⟩⟨1205900
1205908
12059010
⟩ cong ⟨1205900
1205907
12059013
⟩ and ⟨1205902
1205906
⟩ cong ⟨1205905
1205909
⟩(iv) The monoidM contains 6 three-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205907
⟩ cong ⟨1205905
1205908
⟩ ⟨1205902
1205908
⟩ cong ⟨1205905
1205907
⟩ and ⟨1205903
⟩ cong
⟨1205904
⟩
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 7
(v) The monoid M contains 8 four-element monoidseach contains 120590
0
with 4 isomorphism types Theseare semigroups from two preceding items that can besubstantially extended by 120590
0
(vi) The monoid M contains 5 four-element semigroups
not monoids with 3 isomorphism types These are⟨1205906
1205907
⟩ cong ⟨1205908
1205909
⟩ ⟨1205906
1205908
⟩ cong ⟨1205907
1205909
⟩ and ⟨1205907
1205908
⟩These semigroups extended by 120590
0
yield 5 monoids allwhich consist of five elements with 3 isomorphismtypes
(vii) ThemonoidM contains 10 five-element semigroupsmdashnot monoids with 5 isomorphism types These are⟨1205902
12059011
⟩ cong ⟨1205905
12059012
⟩ ⟨1205902
12059012
⟩ cong ⟨1205905
12059011
⟩ ⟨1205902
12059013
⟩ cong
⟨1205905
12059010
⟩ ⟨1205903
1205906
⟩ cong ⟨1205904
1205909
⟩ and ⟨1205903
1205909
⟩ cong ⟨1205904
1205906
⟩These semigroups extended by120590
0
yield 10monoids allof which consist of six elements with 5 isomorphismtypes We get 10 new isomorphism types since thesemigroups are distinguished by semigroups ⟨120590
3
⟩
and ⟨1205904
⟩ and by nonisomorphic semigroups ⟨1205906
1205907
⟩⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(viii) The monoid M contains 5 six-element semigroups
not monoids with 3 isomorphism types These are⟨1205902
1205903
⟩ cong ⟨1205904
1205905
⟩ ⟨1205902
1205904
⟩ cong ⟨1205903
1205905
⟩ and ⟨1205902
1205905
⟩These semigroups extended by 120590
0
yield 5 monoidsall of which consist of seven elements with 3 isomor-phism types We get 6 new isomorphism types sincethe semigroups are distinguished by non isomorphicsemigroups ⟨120590
6
1205907
⟩ ⟨1205906
1205908
⟩ and ⟨1205907
1205908
⟩(ix) ThemonoidM contains no seven-element semigroup
not a monoid no eight-element monoid and theonly semigroup ⟨120590
6
1205909
⟩ with exactly eight elementsand the only monoid ⟨120590
0
1205906
1205909
⟩ with exactly nineelements
(x) The monoid M contains 4 nine-element semigroupsnot monoids with 2 isomorphism types These are⟨1205902
1205909
⟩ cong ⟨1205905
1205906
⟩ and ⟨1205903
1205908
⟩ cong ⟨1205904
1205907
⟩ Thesemigroup ⟨120590
2
1205909
⟩ does not contain a semigroupisomorphic to ⟨120590
3
⟩ hence it is not isomorphic to⟨1205903
1205908
⟩ These semigroups extended by 1205900
yield 4monoids all of which consist of ten elements with 2isomorphism types
(xi) The monoid M contains 6 ten-element semigroupsnot monoids with 4 isomorphism typesThese are ⟨120590
1
1205906
⟩ ⟨1205902
1205903
1205908
⟩ cong ⟨1205904
1205905
1205906
⟩⟨1205902
1205904
1205907
⟩ cong ⟨1205903
1205905
1205906
⟩ and ⟨1205902
1205905
1205906
⟩ Thesesemigroups (except ⟨120590
1
1205906
⟩) extended by 1205900
yield 5monoids all of which consist of ten elements with3 isomorphism types We get 6 new isomorphismtypes since the semigroups are distinguished bynot isomorphic semigroups ⟨120590
2
1205903
⟩ ⟨1205902
1205904
⟩ and⟨1205902
1205905
⟩(xii) The monoid M contains 2 isomorphic semigroups
not monoids which consist of 11 elements that is⟨1205902
1205903
1205905
⟩ cong ⟨1205902
1205904
1205905
⟩These semigroups extendedby 1205900
yield 2 isomorphic monoids which consistof 12 elements The monoid M contains no largersemigroup with the exception of itself
10 Cancellation Rules Motivated by SomeTopological Properties
101 Some Consequences of the Axiom 0 = 0minus So far we usedonly the following relations (above named cancellation rules)1205902
∘ 1205902
= 1205902
1205901
∘ 1205901
= 1205900
1205902
∘ 12059012
= 1205906
and 1205902
∘ 12059013
= 1205907
When one assumes119883 = 0 = 0
minus then
119883 = 1205900
(119883) = 1205902
(119883) = 1205905
(119883) = 1205907
(119883)
= 1205908
(119883) = 12059010
(119883) = 12059013
(119883)
0 = 1205901
(119883) = 1205903
(119883) = 1205904
(119883) = 1205906
(119883)
= 1205909
(119883) = 12059011
(119883) = 12059012
(119883)
(18)
Using the substitution 119860 997891rarr 119860119888 one obtains equiv-
alent relations between operations from the set 1205901
1205903
1205904
1205906
1205909
12059011
12059012
and conversely Therefore cancellationrules are topologically reasonable only between the opera-tions from the monoid as follows
⟨1205900
1205902
1205905
⟩ = 1205900
1205902
1205905
1205907
1205908
12059010
12059013
(19)
Chapman see [3] considered properties of subsets withrespect to such relations Below we are going to identifyrelations that are determined by some topological spacescompare [10 11]
102 The Relation 1205900 = 1205902 If a topological space 119883 isdiscrete then there exist two Kuratowski operation onlyThese are 120590
0
and 1205901
So themonoid of Kuratowski operationsreduced to the group ⟨120590
1
⟩The relation 120590
0
= 1205902
is equivalent to any relation 1205900
=
120590119894
where 119894 isin 5 7 8 10 13 Any such relation implies thatevery subset of 119883 has to be closed and open that is 119883 hasto be discrete However one can check these using (only) thefacts that 120590
1
is an involution and 1205902
is an idempotent and thecancellation rules that is the Cayley table forM So 120590
0
= 1205902
follows
1205900
= 1205902
= 1205905
= 1205907
= 1205908
= 12059010
= 12059013
(20)
103 The Relation 1205902 = 1205905 Topologically 1205902 = 1205905
meansthat 119883 must be discrete This is so because 119860119888minus119888 sube 119860 sube 119860
minus
for any 119860 sube 119883
104 The Relation 1205902 = 1205907 Topologically the relation 1205902 =1205907
implies 1205900
= 1205902
But it requires the use of topology axioms0 = 0minus and 119862minus cup 119861minus = (119862 cup 119861)minus for each 119862 and 119861
Lemma 5 For any topological space 1205902
= 1205907
implies 1205902
= 1205908
Proof If 1205902
= 1205907
then 119860 = 0 rArr 119860119888minus119888
= 0 for any 119860 sube 119883Indeed if 119860119888minus119888 = 0 then 120590
7
(119860) = 0minus
= 0 Since 119860 = 0 then1205902
(119860) = 0 Hence 1205902
(119860) = 1205907
(119860) a contradictionThe axiom 119862
minus
cup 119861minus
= (119862 cup 119861)minus implies that always
(119860minus
cap 119860minus119888minus
)119888minus119888
= 0 (21)
Thus the additional assumption 1205902
= 1205907
follows that always119860minus
cap 119860minus119888minus
= 0 Therefore always 119860minus119888minus = 119860minus119888 but this means
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Mathematics
that any closed set has to be open in other words 1205902
= 1205908
Proposition 6 For any topological space 1205902
= 1205907
implies 1205900
=
1205902
Proof But the relation 1205902
= 1205907
is equivalent with 1205905
= 1205908
ByLemma 5 we get 120590
2
= 1205905
Finally 1205900
= 1205902
The relation 1205902
= 1205907
has interpretation without the axiom0 = 0minus Indeed suppose119883 = 119886 119887 Put
1205902
(0) = 119886 = 1205902
(119886) 119883 = 1205902
(119883) = 1205902
(119887) (22)
Then check that 1205902
= 1205907
and
1205908
(0) = 1205905
(119886) = 1205904
(119887) = 1205901
(119883) = 0 (23)
in other words 1205902
= 1205907
and 1205902
= 1205908
However 1205902
= 1205907
isequivalent to 120590
5
= 1205908
This relation implies 120590
2
= 1205907
= 12059010
1205905
= 1205908
= 12059013
1205903
= 1205906
= 12059011
and 1205904
= 1205909
= 12059012
For this interpretationthe monoidM(120590
2
= 1205907
) consisting of Kuratowski operationover a such 119883 has six elements only In M(120590
2
= 1205907
) thereare relations covered by the following proposition only
Proposition 7 For any monoid with the Cayley table as forM the relation 120590
2
= 1205907
implies
(i) 1205902
= 1205907
= 12059010
= 1205907
∘ 1205902
(ii) 1205905
= 1205901
∘1205902
∘1205901
= 1205901
∘1205907
∘1205901
= 1205908
= 1205905
∘1205902
= 1205905
∘1205907
=
12059013
(iii) 120590
3
= 1205902
∘ 1205901
= 1205907
∘ 1205901
= 1205906
= 12059010
∘ 1205901
= 12059011
(iv) 1205904
= 1205901
∘ 1205902
= 1205901
∘ 1205907
= 1205909
= 1205901
∘ 12059010
= 12059012
Thus the Cayley table does not contain the completeinformation resulting from the axioms of topology
105 The Relation 1205902 = 1205908 Topologically the relation 1205902 =1205908
means that any closed set is open tooThus if119883 = 119886 119887 119888
is a topological space with the open sets 119883 0 119886 119887 and 119888thenM(120590
2
= 1205908
) is the monoid of all Kuratowski operationsover 119883 Relations 120590
2
= 1205908
and 1205905
= 1205907
are equivalent Theyimply relations 120590
2
= 1205908
= 12059010
1205905
= 1205907
= 12059013
1205903
= 1205909
= 12059011
and 120590
4
= 1205906
= 12059012
The permutation
(1205900
1205901
1205902
1205903
1205904
1205905
1205900
1205901
1205905
1205904
1205903
1205902
) (24)
determines the isomorphism between monoidsM(1205902
= 1205907
)
andM(1205902
= 1205908
)
106 The Relations 1205902 = 12059010 and 1205902 = 12059013 Topologicallythe relation 120590
2
= 1205908
means that any nonempty closed set hasnonempty interior For each 119860 the closed set 119860minus cap 119860minus119888minus hasempty interior so the relation 120590
2
= 12059010
implies that 119860minus119888 isclosed Hence any open set has to be closed so it implies1205902
= 1205908
The relation 1205902
= 12059013
follows that each closed sethas to be open so it implies 120590
2
= 1205908
too
107The Relation 1205907 = 1205908 Using the Cayley table forM onecan check that the relations 120590
7
= 1205908
and 12059010
= 12059013
areequivalent Each of them gives 120590
7
= 1205908
= 12059010
= 12059013
and1205906
= 1205909
= 12059011
= 12059012
If 119883 = 119886 119887 is a topological space withthe open sets119883 0 and 119886 thenM(120590
7
= 1205908
) is themonoid ofall Kuratowski operations over119883 and consists of 8 elements
108 The Relation 1205907 = 12059010 Using the Cayley table for Mone can check that the relations 120590
7
= 12059010
1205908
= 12059013
1205906
= 12059011
and 120590
9
= 12059012
are equivalent If 119883 is a sequence converging tothe point 119892 and 119892 isin 119883 thenM(120590
7
= 12059010
) is the monoid of allKuratowski operations over119883 and consists of 10 elements
109 The Relation 1205907 = 12059013 Using the Cayley table forMone can check that the relations 120590
7
= 12059013
and 1205908
= 12059010
areequivalent Also 120590
6
= 12059012
and 1205909
= 12059011
These relations givethe monoid with the following Cayley table where the rowand column marked by the identity are omitted
1205901
1205902
1205903
1205904
1205905
1205906
1205907
1205908
1205909
1205901
1205900
1205904
1205905
1205902
1205903
1205908
1205909
1205906
1205907
1205902
1205903
1205902
1205903
1205906
1205907
1205906
1205907
1205908
1205909
1205903
1205902
1205906
1205907
1205902
1205903
1205908
1205909
1205906
1205907
1205904
1205905
1205904
1205905
1205908
1205909
1205908
1205909
1205906
1205907
1205905
1205904
1205908
1205909
1205904
1205905
1205906
1205907
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
1205907
1205906
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205908
1205909
1205908
1205909
1205906
1205907
1205906
1205907
1205908
1205909
1205909
1205908
1205906
1205907
1205908
1205909
1205908
1205909
1205906
1205907
(25)
If a space119883 is extremally disconnected then the closuresof open sets are open compare [2 page 452] It follows that1205906
= 12059012
The space 119883 = 119886 119887 with the open sets 119883 0and 119886 is extremally disconnected But it contains a one-element open and dense set 119886 and it follows that 120590
7
= 1205908
Similar is for the space 120573119873 see [2 pages 228 and 453] tofind the definition and properties of 120573119873 There are Hausdorffextremally disconnected spaces which are dense in itself Forexample the Stone space of the complete Boolean algebra ofall regular closed subsets of the unit interval compare [12]For such spaces 120590
7
= 1205908
and 1205907
= 12059013
To see this suppose aHausdorff 119883 is extremally disconnected and dense in itselfLet 119883 = 119880 cup 119881 cup 119882 where sets 119880 119881 and119882 are closed andopen Consider a set 119860 = 119860119888minus119888 cup 119861 cup 119862 such that
(i) 119862119888minus119888 = 0 and 119862minus = 119882(ii) 0 = 119861 sube 119881 and 119861minus119888minus119888 = 0(iii) 119880 = 119860119888minus119888minus = 119860
119888minus119888Then check that
(i) 1205900
(119860) = 119860 and 1205901
(119860) = 119883 (119860119888minus119888
cup 119861 cup 119862)(ii) 1205902
(119860) = 119880 cup 119861minus
cup119882 and 1205903
(119860) = 119883 119860119888minus119888
(iii) 1205904
(119860) = 119881 119861minus and 120590
5
(119860) = 119860119888minus119888
(iv) 1205906
(119860) = 12059012
(119860) = 119881 and 1205907
(119860) = 12059013
(119860) = 119880(v) 1205908
(119860) = 12059010
(119860) = 119880 cup 119882 and 1205909
(119860) = 12059011
(119860) =
119881 cup 119882Hence we have that 120590
7
= 1205908
Note that if119882 = 0 then 1205907
(119860) =
1205908
(119860) This is the case of subsets of 120573119873
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 9
References
[1] K Kuratowski ldquoSur lrsquooperation 119860 de lrsquoanalysis situsrdquo Funda-menta Mathematicae vol 3 pp 182ndash199 1922
[2] R EngelkingGeneral Topology PWN-Polish Scientific Publish-ers Warsaw Poland 1977
[3] T A Chapman ldquoA further note on closure and interioroperatorsrdquoThe American Mathematical Monthly vol 69 no 6pp 524ndash529 1962
[4] D Sherman ldquoVariations onKuratowskirsquos 14-set theoremrdquoAmer-ican Mathematical Monthly vol 117 no 2 pp 113ndash123 2010
[5] B J Gardner and M Jackson ldquoThe Kuratowski closure-complement theoremrdquo New Zealand Journal of Mathematicsvol 38 pp 9ndash44 2008
[6] W Koenen ldquoThe Kuratowski closure problem in the topologyof convexityrdquoThe American Mathematical Monthly vol 73 pp704ndash708 1966
[7] K-P Shum ldquoClosure functions on the set of positive integersrdquoScience in China Series A vol 39 no 4 pp 337ndash346 1996
[8] W Sierpinski General Topology Mathematical ExpositionsUniversity of Toronto Press Toronto Canada 1956
[9] A Cayley ldquoOn the theory of groupsrdquo American Journal ofMathematics vol 11 no 2 pp 139ndash157 1889
[10] C E Aull ldquoClassification of topological spacesrdquo Bulletinde lrsquoAcademie Polonaise des Sciences Serie des SciencesMathematiques Astronomiques et Physiques vol 15 pp773ndash778 1967
[11] C E Aull ldquoCorrigendumml ldquoClassification of topologicalspacesrdquordquo Bulletin de lrsquoAcademie Polonaise des Sciences Serie desSciencesMathematiques Astronomiques et Physiques vol 16 no6 1968
[12] A M Gleason ldquoProjective topological spacesrdquo Illinois Journalof Mathematics vol 2 pp 482ndash489 1958
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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