SU · 2010. 3. 12. · พีชคณิัตอันดบพรีมอลแบบเทม โดย นางสาวรัตนา ศรีทั ศน วิทยานิ

พชคณตอนดบพรมอลแบบเทม

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นางสาวรตนา ศรทศน

วทยานพนธนเปนสวนหนงของการศกษาตามหลกสตรปรญญาวทยาศาสตรมหาบณฑต สาขาวชาคณตศาสตร ภาควชาคณตศาสตร

บณฑตวทยาลย มหาวทยาลยศลปากร ปการศกษา 2545

ISBN 974-653-261-8

ลขสทธของบณฑตวทยาลย มหาวทยาลยศลปากร

TAME ORDER-PRIMAL ALGEBRAS

By

Ratana Srithus

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree

MASTER OF SCIENCE

Department of Mathematics

Graduate School

SILPAKORN UNIVERSITY

2002

ISBN 974-653-261-8

บณฑตวทยาลย มหาวทยาลยศลปากร อนมตใหวทยานพนธเรอง “พชคณตอนดบพรมอลแบบเทม” เสนอโดย นางสาวรตนา ศรทศน เปนสวนหนงของการศกษาตามหลกสตรปรญญาวทยาศาสตรมหาบณฑต สาขาวชาคณตศาสตร

………………………………………………...

(ผชวยศาสตราจารย ดร. จราวรรณ คงคลาย) คณบดบณฑตวทยาลย วนท………..เดอน…………..พ.ศ……………. ผควบคมวทยานพนธ

รองศาสตราจารย ดร. ฉววรรณ รตนประเสรฐ

คณะกรรมการตรวจสอบวทยานพนธ ………………………………..ประธานกรรมการ (รองศาสตราจารย ดร.นวรตน อนนตชน) ………../……………./…………. ………………………………..กรรมการ (รองศาสตราจารย ดร.ฉววรรณ รตนประเสรฐ) ………../……………./…………. ………………………………..กรรมการ (รองศาสตราจารย ดร.ณรงค ปนนม) ………../……………./………….

i

K43511004 : สาขาวชาคณตศาสตร คาสาคญ : พชคณตเลกสดเฉพาะกลม / เทม/ พชคณตอนดบพรมอลแบบเทม รตนา ศรทศน : พชคณตอนดบพรมอลแบบเทม (TAME ORDER-PRIMAL ALGEBRAS) อาจารยผควบคมวทยานพนธ : รศ. ดร. ฉววรรณ รตนประเสรฐ. 58 หนา. ISBN 974-653-261-8

กาหนดให h : A → B เปนฟงกชนถายแบบไมคงตว แลวเปนททราบดวา h(A) เปนพชคณตยอยของ B ซงมขนาดมากกวาหรอเทากบ 2 และ ถา A เปนพชคณตแบบเทม เราสามารถแสดงเงอนไขจาเปนและเพยงพอทจะทาใหอมเมจของฟงกชนถายแบบทงหมดของ A เปนพชคณตแบบเทม พชคณตอนดบพรมอล A ซงขนอยกบเซตอนดบ ( A; ≤ ) คอพชคณตซง T(A) = Pol(≤ ) ใน [3] ไดพสจนวาถา ( A; ≤ ) เปนปฏโซหรอเซตอนดบตอเนองแลว A เปนพชคณตอยางงาย เราพสจนวา ปฏโซหรอเซตอนดบตอเนองเปนเงอนไขจาเปนและเพยงพอสาหรบการเปนพชคณตแบบเทม นอกจากนเรายงพสจนวาทกพชคณตอยางงายเปนพชคณตแบบเทม ทาใหเราจาแนกพชคณตอนดบพรมอลทงหมดทเปนเทม นนคอพสจนวา พชคณตอนดบพรมอล A เปนเทม กตอเมอ ( A; ≤ ) เปนปฏโซหรอเซตอนดบตอเนอง ยงไปกวานนเราพสจนไดวา แตละพชคณตอนดบพรมอลเปนนอนอาบเลยนและมเซตเลกสดเฉพาะกลมทมขนาดเทากบ 2

ในทฤษฎคอนกรเอนซแบบเทม ไดมการจาแนก type ของพชคณตเลกสดเฉพาะกลมทงหมดออกเปน type 1-5 และแตละพชคณตแบบเทมจะมอยางนอยหนง type ซงแตละ type ของมนจะคอ type ของพชคณตเลกสดเฉพาะกลมของตวเอง ใน [5] ไดมการจาแนกพชคณตทงหมดทเปนอาบเลยนโดยใชคณสมบตของ type ของพชคณต นนคอ พชคณต A เปนอาบเลยน กตอเมอ type A ∈{1,2} ทาใหเราไดวา แตละพชคณตอนดบพรมอล A ไมใช type 1 หรอ 2 หรอกคอ type A ⊆ {3,4,5} เราจงจาแนก type ทงหมดของพชคณตอนดบพรมอลแบบเทม A โดยใชคณสมบตของเซตอนดบ ( A; ≤ ) ทขนอยกบ A นนคอพสจนวา ( A; ≤ ) เปนปฏโซ กตอเมอ type { A } = {3} และ ( A; ≤ ) เปนเซตอนดบตอเนอง กตอเมอ type { A } = {4} นอกจากนเรายงสามารถหา type ทเปนไปไดทงหมดของวาไรตทกอกาเนดโดยพชคณตอนดบพรมอลแบบเทม A และใหเงอนไขเพยงพอททาให V(A) เปน residually small

ภาควชาคณตศาสตร บณฑตวทยาลย มหาวทยาลยศลปากร ปการศกษา 2545 ลายมอชอนกศกษา…………………………………….. ลายมอชออาจารยผควบคมวทยานพนธ………………………………………

K43511004 : MAJOR : MATHEMATICS

KEY WORD : MINIMAL ALGEBRA/TAME/ORDER-PRIMAL ALGEBRA

RATANA SRITHUS : TAME ORDER-PRIMAL ALGEBRAS. THESIS

ADVISOR : ASSO. PROF. CHAWEEWAN RATANAPRASERT, Ph.D. 58 pp. ISBN974-653-261-8.

Let h:A → B be a non-constant homomorphism. It is well-known that h(A) is asubalgebra of B whose cardinality is greater than 2. If A is a tame algebra, we give a necessaryand sufficient condition that all images of A under homomorphisms are tame.

An order-primal algebra A corresponding to an ordered set (A;≤) is an algebrasuch that T (A) = Pol(A). It was shown in [3] that if (A;≤) is either an antichain or connected,then A is simple. We can prove that an antichain or a connected property is a necessary andsufficient condition for A to be tame. Moreover, we can prove that every simple algebra istame. By these results, we can characterize all order-primal algebras which are tame; that is,an order-primal algebra A is tame if and only if (A;≤) is either an antichain or connected.Furthermore, we can prove that every order-primal algebra is non-abelian and it has a minimalset whose cardinality is 2.

In Tame Congruence Theorey, they characterized all minimal algebras by assign-ing type 1-5 and each tame algebra A has at least one type which is the same type of itsminimal algebra. In [5], they characterize all algebras to be abelian by using propertiesof types of an algebra; that is, an algebra A is abelian if and only if type A ∈ {1, 2}.By this result, every type of an order-primal algebra is not type 1 or 2 which impliesthat type {A} ⊆ {3, 4, 5}. We can characterize all types of a tame order-primal alge-bra A by the property of the ordered set (A;≤) corresponding to A; that is, (A;≤) isan antichain if and only if type {A} = {3} and (A;≤) is connected if and only iftype {A} = {4}. Moreover, we can find all possible types of a variety generated by anorder-primal algebra A and give some sufficient conditions that V (A) is residually small.

Department of Mathematics Graduate School, Silpakorn University Academic Year 2002

Student’s signature...............................

Thesis Advisor’s signature..........................

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Acknowledgements

This thesis has been completed by the involvement of many people about whom Iwould like to mention here.

I would like to express my gratitude and sincere appreciation to Asso. Prof. Dr.Chaweewan Rattanaprasert, my advisor for her valuable suggestions and excellent advicesthroughout the study with great attention.

I would like to thank Asso. Prof. Dr. Nawarat Ananchuen and Asso. Prof. Dr.Narong Punnim, Chairman and Member of the thesis Committee, for their valuable commentsand suggestions.

Finally, I would like to express my gratitude to my family and my friends for theirunderstanding, encouragement and moral support during the study.

Ratana Srithus

April 2003

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Contents

page .

Abstract in Thai.................................................................................................iv

Abstract in English..............................................................................................v

Acknowledgements..............................................................................................vi

List of Table......................................................................................................viii

Introduction..........................................................................................................1

Chapter 1 : Basic Concepts of Universal Algebras................................................4

Chapter 2 : Minimal Algebras.............................................................................19

Chapter 3 : Some Properties of Tame Algebras...................................................27

Chapter 4 : Types of Order-primal Algebras........................................................39

References..............................................................................................................55

Appendix A : Lattice Theory..............................................................................56

Biography..............................................................................................................58

iii

List of Tables

Table page .

1.1 Definitions of projections............................................................................5

1.2 Definition of operation ·..............................................................................72.1 Definitions of m1 and m2...........................................................................22

2.2 Definitions of f1, . . . , f6...............................................................................22

3.1 Definitions of f and g..................................................................................37

4.1 Definition of t ............................................................................................44

4.2 Definition of p ............................................................................................44

4.3 Definition of t .............................................................................................47

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1

Introduction

An algebra A composes a pair consisting of a nonempty set Awhich together with a set of finitary operations FA defined on A. A set A iscalled a universe and every element of FA is called a fundamental operation ofA.

In Chapter 1, we will give some important basic concepts in UniversalAlgebras which will be used in this thesis.

Let θ be an equivalence relation on A. Then θ is called congruencerelation on an algebra A if θ satisfies the compatibility property. We denote theset of all congruence relations of A by ConA. One can show that ConA is acomplete lattice.

Let U ⊆ A. We define by P (A)|U the set of all restrictions of polyno-mial operations f of A to U in which U is closed with respect to f . An inducedalgebra of A by a subset U of A, denoted by A|U , is an algebra in which U is anuniverse and P (A)|U is the set of all fundamental operations of A|U . A mappinge: A → A is called idempotent if e ◦ e = e.

It was shown in [5] that the restriction congruence relation of an al-gebra A to an image of an idempotent polynomial operation U of A is also acongruence relation on the induced algebra A|U . In Chapter 2, we introduce theproperty of the congruence lattice of an algebra A which force itself to be tame.This property involves to be tight. By using the result in [5], we can prove thatif ConA is tight then ConA|U is also tight.

An algebra A is called finite if A is finite. Finite algebras are importantin many areas where finiteness plays a crucial role, especially in computer science.At the beginning of the eighties R. McKenzie and D. Hobby developed a newtheory, is called ”Tame Congruence Theory” which is a structure theory forfinite algebras. An important concept of this theory is the notion of a minimalalgebra.

Let U be a subset of a set A. Then U is a minimal set of an algebra A ifU is the image of a unary polynomial operation of A and U is a minimal elementof the set of all images of unary polynomial operations of A whose cardinalitiesare greater than 1. An induced algebra A|U is called a minimal algebra if U is aminimal set.

An algebra A is called tame if A has a minimal set U in which U is animage of an idempotent operation of A and every restriction of nontrivial propercongruence relation of A to U is also a nontrivial proper congruence relation ofa minimal algebra A|U .

2

Let h: A → B be a homomorphism. It is well-known that h(A) is asubalgebra of B. If A is tame, we want that h(A) is tame. Obviously, if h isconstant, then A is not tame. So, we will find some sufficient conditions for h(A)to be tame.

K. Denecke [5] showed that all minimal sets of a tame algebra satisfythe tame conditions and all minimal algebras of a tame algebra are isomorphicand gave a sufficient condition for algebras to be tame. So, we will considerthe similar results for induced algebras. In Chapter 3, we can characterize allhomomorphic images of a tame algebra in which h(A) is tame and we give somesufficient conditions for induced algebras to be tame.

A clone C on a set A is a set of operations defined on A which is closedunder composition and contains all projections. If A = (A; FA) is an algebrathen the clone of all term operations T (A) of A is the clone which is generatedby FA. Let O(A) be the clone of all operations defined on A. A finite algebraA = (A; FA) is called primal if T (A) = O(A); that is, every operation definedon A is a term operation of A.

Let ≤ be a binary relation on A. Then ≤ is called order on a set Aif ≤ is reflexive, anti-symmetric and transitive. For an order ≤, we often preferto write x ≤ y in place of (x, y) ∈≤. Given an order ≤ on a nonempty set A,the pair (A;≤) is called an ordered set. Let (A;≤) be a finite ordered set andlet Pol(≤) be the set of all operations defined on A which preserve the order≤. Then a finite algebra A = (A; FA) is called order-primal if T (A) = Pol(≤).An ordered set (A;≤) is called connected if for any two elements a, b ∈ Athere exist a natural number n and elements a = a0, a1, . . . , an = b such thata0 ≤ a1 ≥ a2 ≤ . . . ≥ an(≤ an) = b (or a0 ≥ a1 ≤ a2 ≥ . . . ≤ an(≥ an) = b).Clearly, if (A;≤) has the least element or the greatest element then (A;≤) isconnected. An ordered set (A;≤) is called an antichain if for any two elementsa, b ∈ A are non-comparable.

An abelian algebra A is an algebra which satisfies the term con-dition if for all n ∈ N, every n-ary term operation tA of A and alla, b, c2, . . . , cn, d2, . . . , dn ∈ A the following implication is satisfied

tA(a, c2, . . . , cn) = tA(a, d2, . . . , dn) ⇒ tA(b, c2, . . . , cn) = tA(b, d2, . . . , dn)

In Tame Congruence Theory, one can characterize all minimal algebrasto be type 1-5 and every tame algebra has at least one type in which every itstype is a type of a minimal algebra of the algebra. In [5], they chracterized allabelian algebras by using properties of types of the algebra; that is, an algebraA is abelian if and only if type {A} = {1, 2}.

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K. Denecke [5] gave a classification of all minimal algebras by usingthe properties of the congruence lattices of all algebras in a variety generatedby a minimal algebra.

In Tame Congruence Theory, the definition of type can be extened tovarieties; that is, every type of a variety of algebras is a type of a finite algebrain the variety.

D. Hobby and R. McKenzie [5] gave a characterization of all typeswhich can occur for a variety by means of the Mal’cev-condition properties ofthe variety.

Let V be a variety. We denote by κ(V ) the least cardinal number λsuch that every subdirectly irreducible algebra in V has the cardinality less thanλ. If there is such a cardinal number; in this case we say that V is residuallysmall. If no such cardinal number exist, we let κ(V ) = ∞ and is said to beresidually large.

In was shown in [3] that if (A;≤) is either an antichain or connected,then A is simple. In Chapter 4, we can prove that an antichain or a connectedproperty is a necessary and sufficient condition for A to be tame. Moreover, wecan prove that every simple algebra is tame. By these result, we can characterizeall order-primal algebras which is tame; that is, an order-primal algebra A istame if and only if (A;≤) is either an antichain or connected. So, we can studysome properties of order-primal algebras A via Tame Congruence Theory, i.e wecan find all types of A and all possiable type of variety generated by A via TameCongruence Theory. And we can prove first that every order-primal algebra isnon-abelian and it has a minimal set whose cardinality is 2. From this result, weget that every type of an order-primal algebra is not type 1 or 2 which impliesthat type {A} ⊆ {3, 4, 5}. Moreover, we can characterize all types of tameorder-primal algebras A by using the K. Denecke results and the properties ofthe ordered set (A;≤) corresponding to A; that is, (A;≤) is an antichain ifand only if type {A} = {3}; and (A;≤) is connected if and only if type{A} = {4} and we can find some sufficient conditions for a variety generatedby an order-primal algebra to be join-semidistutive. From these result, we canfind all possible types of a variety generated by an order-primal algebra. For avariety generated by an order-primal algebra, we give some sufficient conditionsfor a variety generated by an order-primal algebra to be residually small.

Chapter 1

Basic Concepts of UniversalAlgebras

In this chapter, we will give some important basic concepts in UniversalAlgebras which we will refer in sequel.

1.1 Definitions and Examples of Algebras

Let A be a nonempty set. We define A0 = {∅} and for a natural numbern, we let An be the set of all n-tuples of elements of A. An n-ary operation fon A is an operation from An into A and we call n the arity of f . A finitaryoperation f on A is a n-ary operation for some natural number n. The imageof (a1, . . . , an) under a n-ary operation f is denoted by f(a1, . . . , an). Becauseevery operation taken up in this thesis will be finitary, we will generally omitthe word ”finitary” and use ”operation” to mean finitary operation. If A isnonempty, then each operation on A has a unique arity. An operation f on Ais called nullary if its arity is zero; it is completely determined by the imagef(∅) in A of the only element ∅ in A0. An operation f on A is unary, binaryor ternary if its arity is 1, 2 or 3; respectively. Let On(A) denote the set of alln-ary operations defined on A and we define O(A) :=

⋃∞n=1 On(A) to be the set

of all operations on A.

For n ≥ 1 and 1 ≤ j ≤ n, the n-ary operation en,Aj defined on A by

en,Aj (a1, . . . , an) = aj is called the j-th projection operation of arity n. Given

a ∈ A, the unary operation ca, which is defined on A by ca(x) = a for all x ∈ A,is called a constant operation of A.

1.1 Example For A = {0, 1} and n = 2, we have two binary projec-tions which are given by Tables 1.1

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5

en,Aj 0 00 0 01 1 1

en,Aj 0 00 0 01 1 1

Table 1.1

¯Let f ∈ Om(A) and g1, . . . , gm ∈ On(A), we define a new n-ary opera-

tion f(g1, ..., gk): Om(A)× (On(A))m → On(A) as follow:

f(g1, ..., gk)(a1, . . . , an) = f(g1(a1, . . . , an), ..., gk(a1, . . . , an))

for all (a1, . . . , an) ∈ An and it is called composition or supercomposition. Aclone on a set A is a set of operations on A which is closed under compositionand contains all the projections en,A

i for all n and all 1 ≤ i ≤ n. Clearly, O(A)is a clone. Given H ⊆ O(A), we denote

< H >=⋂{C|H ⊆ C and C is a clone of A }

to be the smallest clone containing H and we call it ”the clone generated by H”.

A non-indexed algebra A composes of a pair consisting of a nonemptyset A together with a set of operations FA defined on A. A is called the universeof A and every element f of F is called a fundamental operation of A.

An indexed algebra A is an ordered pair (A; (fAi )i∈I) such that A is a

nonempty universe and for each i, fi, which is called a fundamental operationof A, is a ni-ary operation on A for some natural number ni and I is called theindexed set.

The sequence τ := (ni)i∈I of arities of all fundamental operations of Ais called the type of A. We denote the class of all algebras of type τ by Alg(τ).An algebra A is finite if its universe A is finite.

1.2 Example The followings are important examples of algebras.

(1) A semilattice S = (S; ·) is an algebra of type (2) such that S satisfies, forall x, y, z ∈ S

x · (y · z) = (x · y) · zx · x = x.

(2) A lattice L = (L;∧,∨) is an algebra with two binary operations ∧ and∨, which are called meet and join; respectively, satisfying the followingidentities :

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x ∨ y = y ∨ x and x ∧ y = y ∧ x(x ∨ y) ∨ z = x ∨ (y ∨ z) and (x ∧ y) ∧ z = x ∧ (y ∧ z).x ∨ x = x and x ∧ x = xx = x ∨ (x ∨ y) and x = x ∧ (x ∧ y).

for all x, y, z ∈ L.

A lattice L is called distributive lattice if it satisfies the distributive law

x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

for all x, y, z ∈ L.

A lattice L is called modular if it satisfies the modular law

x ≤ y ⇒ x ∧ (y ∨ z) = (x ∧ y) ∨ z for all x, y, z ∈ L.

(3) A bounded lattice L = (L;∧,∨, 0, 1) is an algebra of type (2,2,0,0) suchthat (L;∧,∨) is a lattice and

x ∨ 1 = 1 and x ∧ 0 = 0 for all x ∈ L.

(4) A Boolean algebra B = (B;∧,∨, ′, 0, 1) is an algebra of type (2,2,1,0,0)such that (B;∧,∨, 0, 1) is a bounded distributive lattice and

x ∨ x′ = 1 and x ∧ x′ = 0 for all x ∈ B.

(5) Let R = (R; +, ·,−, 0) be a ring. A R-module M = (M ; +,−, 0, R) isan algebra such that (M ; +,−, 0) is an abelian group and the followingidentities are satisfied for all r, s ∈ R and x, y ∈ M ,

r(x + y) = r(x) + r(y)

(r + s)x = r(x) + s(x)

(r · s)x = r(s(x)).

In the special case when R is a field, a R-module is a R-vector space. Everyelement of R-vector space is called vector and every field element is calledscalar. ¯

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1.2 Subalgebras, Homomorphisms and Isomor-

phisms

The concept of subalgebra is an important method of constructing newalgebras from a given one.

1.3 Definition Let A and B be two algebras of the same type. Then Ais called a subalgebra of B if A satisfies the following conditions

(i) A ⊆ B

(ii) every fundamental operation of A is the restriction of the cor-responding operation of B.

We write A ≤ B for A is a subalgebra of B.

The following lemma, which was proved in [5], gives a criterion for asubset of the universe of an algebra to be a subalgebra.

1.4 Lemma (Subalgebra Criterion)[5] Let B = (B; (fBi )i∈I) be an al-

gebra of type τ and let A ⊆ B and fAi = fB

i|Afor all i ∈ I. Then A = (A; (fA

i )i∈I)

is a subalgebra of B if and only if fBi (Ani) ⊆ A for all i ∈ I.

1.5 Example Consider the group G = ({1,−1, i,−i}; ·) where · isdefined as Table 1.2

· 1 -1 i -i1 1 -1 i -i-1 -1 1 -i ii i -i -1 1-i -i i 1 -1.

Table 1.2

Then G is an abelian group with 1 being the identity element. One can see that({1}; ·) and ({1,−1}; ·) are subalgebras of G but ({1,−i}; ·) and ({1,−1,−i}; ·)are not subalgebras of G. ¯

1.6 Definition Let A = (A; (fAi )i∈I) and B = (B; (fB

i )i∈I) be algebrasof the same type. Then a mapping h : A → B is called a homomorphism fromA into B if

h(fAi (a1, . . . , ani

)) = fBi (h(a1), . . . , h(ani

))

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for all a1, . . . , ani∈ A. If h is surjective, then B is called a homomorphic

image of A and h is called an epimorphism. A homomorphism h is called amonomorphism or embedding if h is injective. A homomorphism h is called anisomorphism if h is bijective. A homomorphism of an algebra A into itself iscalled an endomorphism of A and an isomorphism from an algebra A into itselfis called an automorphism of A.

1.7 Example It is clear that the identity mapping idA: A → A whichis defined by idA(x) = x for all x ∈ A is an automorphism of every algebra A.¯

The following theorems were proved in [1].

1.8 Theorem [1] Let A,B and C be algebras of the same type and letα: A → B and β: B → C be homomorphisms . Then β ◦ α is a homomorphismfrom A into C.

1.9 Theorem [1] Let A and B be algebras of the same type and let h: A →B be a homomorphism. Then the image of a subalgebra of A under h is asubalgebra of B.

1.3 Congruence Relations and Quotient Alge-

bras

Recall that a h-ary relation ρ on a set A is a subset of Ah. If h = 2,we call ρ a binary relation. We denote

aρb ⇐⇒ (a, b) ∈ ρ .

1.10 Definition Let r1 and r2 be binary relations on A. The relationalproduction of r1 and r2, which is written by r1 ◦ r2, is the binary relation on Adefined by

(a, b) ∈ r1 ◦ r2 ⇐⇒ (∃c ∈ A)((a, c) ∈ r2 and (c, b) ∈ r1).

We can inductively define r1 ◦ . . . ◦ rn = (r1 ◦ . . . ◦ rn−1) ◦ rn for all n ≥ 2. Thediagonal relation ∆A on A is the set {(a, a)|a ∈ A} and we denote the universalrelation A2 by ∇A. A binary relation r on A is an equivalence relation on Aif r is reflexive, symmetric and transitive. We denote by Eq(A) the set of allequivalence relations on A.

For θ1 and θ2 ∈ Eq(A). One can show that θ1 ∩ θ2 ∈ Eq(A). Butin general, the union of two equivalence relations on A is not an equivalencerelation, as the following example shows, let

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A = {1, 2, 3}, θ1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} and

θ2 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}.

Clearly, θ1 and θ2 are equivalence relations, but

θ1 ∪ θ2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

is not an equivalence relation on A since it is not transitive.

Given θ ⊆ A× A, we denote

< θ >=⋂{β | θ ⊆ β and β is a congruence relation on A }

to be the smallest congruence relation containing θ and we call it ”the congruencerelation generated by θ”.

The following theorem were proved in [1].

1.11 Theorem [1] Let A be a nonempty set. Then Eq(A) is a completelattice under the inclusion. Moreover, the operations meet and join of Eq(A)are defined as follow:

θ1 ∧ θ2 = θ1 ∩ θ2 θ1 ∨ θ2 =< θ1 ∪ θ2 > for all θ1, θ2 ∈ Eq(A).

1.12 Theorem [1] If θ1 and θ2 are two equivalence relations on A. Then

θ1 ∨ θ2 = θ1 ∪ (θ1 ◦ θ2) ∪ (θ1 ◦ θ2 ◦ θ1) ∪ . . .

or equivalenty, (a, b) ∈ θ1 ∨ θ2 if and only if there is a sequence of elementsc1, . . . , cn ∈ A such that (ci, ci+1) ∈ θ1 or (ci, ci+1) ∈ θ2 for all i = 1, . . . , n − 1and a = c1, b = cn.

1.13 Definition Let ρ be a h-ary relation on the set A and let f be an-ary operation on A. We say that f preserves ρ or f is compatible with ρ if

(f(a11, . . . , an1), . . . , f(a1h, . . . , anh)) ∈ ρ

whenever (a11, . . . , a1h), . . . , (an1, . . . , anh) ∈ ρ. We denote by Pol(ρ) the set ofall operations on A which preserve ρ.

It is well-known that Polρ is a clone.

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1.14 Definition Let A be an algebra. An equivalence relation θ onA is called a congruence relation of A if every fundamental operation fA iscompatible with θ. We denote the set of all congruence relations of A by Con(A).

It is clear that the diagonal ∆A and the universal ∇A are congruencerelations of A. So, {∆A,∇A} ⊆ Con(A). An algebra A is called simple ifCon(A) = {∆A,∇A}.

It is well-known that every mapping h: A → B from a set A into a setB defines an equivalence relation on A by {(a, b) ∈ A2|h(a) = h(b)} and we callthe relation, kernel of h and denoted it by kerh; so,

kerh = {(a, b) ∈ A2|h(a) = h(b)}.

The following lemma and theorem were proved in [1].

1.15 Lemma [1] Let A and B be algebras of the same type and leth: A → B be a homomorphism. Then kerh is a congruence relation on A.

1.16 Theorem [1] (Con(A);⊆) is a complete sublattice of EqA.

Let θ be a congruence relation on an algebra A. We define the followingnotation :

a/θ = {b ∈ A | (a, b) ∈ θ} for each a ∈ A

A/θ = {a/θ | a ∈ A}

For a ∈ A, the set a/θ is called the congruence class of θ and the set of allcongruence classes of θ A/θ is the partition of A which is called the quotient set.

Each congruence relation θ on an algebra can induce a new algebrafrom a given algebra as follows.

1.17 Definition Let A be an algebra and let θ ∈ Con(A). Then thequotient algebra of A by θ, which is denoted by A/θ, is the algebra whoseuniverse is the quotient set A/θ. For each i, there is a ni-ary fundamental

operation fAi of A and we define a ni-ary operation f

A/θi on A/θ corresponding

to fAi by

fA/θi (a1/θ, . . . , ani

/θ) = fAi (a1, . . . , ani

)/θ

for all a1, . . . , ani∈ A.

The following theorems are important in Universal Algebras. One canfind in every book about Universal Algebras. Here, we quote from [1].

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1.18 Theorem [1] Let A be an algebra and let θ ∈ Con(A). Then anatural map η: A → A/kerϕ, which is defined by η(a) = a/θ for all a ∈ A, is anepimorphism.

1.19 Theorem (Homomorphism Theorem)[1] Let A and B be algebrasand let ϕ: A → B be an epimorphism. Then there is an isomorphism β fromA/kerϕ to B which is defined by ϕ = β◦η where η is the natural homomorphismfrom A to A/kerϕ.

In 1963 G. Grazer and C. Schmidt proved in [1] that for every algebraiclattice L, there is an algebra A such that L ∼= Con(A). One can ask that underwhat conditions on A imply Con(A) to be distributive or modular. To look forthe answer, we need the following concepts.

1.20 Definition An algebra A is congruence distributive or congruencemodular if Con(A) is a distributive or modular lattice.

If θ1, θ2 ∈ Con(A) and θ1 ◦ θ2 = θ2 ◦ θ1, we say that θ1 and θ2 arepermutable; or equivalently θ1 and θ2 are permute. An algebra A is n-permutableif Con(A) satisfies the n-permutable property; that is, for all θ, β ∈ Con(A) wehave

θ ∨ β = θ1 ◦ . . . ◦ θn where θi =

{θ, i is evenβ, i is odd

An algebra A is called meet-semidistributive if all θ1, θ2, θ3 ∈ Con(A) satisfy theimplication

θ1 ∩ θ2 = θ1 ∩ θ3 ⇒ θ1 ∩ θ2 = θ1 ∩ (θ2 ∪ θ3).

Dually, A is called join-semidistributive if all θ1, θ2, θ3 ∈ Con(A) satisfy theimplication

θ1 ∪ θ2 = θ1 ∪ θ3 ⇒ θ1 ∪ θ2 = θ1 ∪ (θ2 ∩ θ3).

An algebra A is called semidistributive if A is both meet-semidistributive andjoin-semidistributive.

1.4 Direct Products, Subdirect Products and

Varieties

The formation of product of algebras is one of the most importantconstructions. One can lead to new algebra with bigger cardinality than thosewe started.

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1.21 Definition Let AJ = (Aj)j∈J be an indexed family of algebrasof type τ . The direct product of AJ , which is written by Πj∈JAj, is the algebrawhose universe is the set

Πj∈JAj := {(aj)j∈J |aj ∈ Aj for all j ∈ J}

where (aj)j∈J is a function from J into⋃

j∈J Aj defined by (aj)j∈J(i) = ai ∈ Ai

and for each i ∈ I, there are ni-ary operations fAJi of Aj for all j ∈ J and we

define a ni-ary operation fΠj∈JAj

i on the direct product corresponding to fAJi for

all j ∈ J by

fΠj∈JAj

i ((a1j)j∈J , . . . , (anij)j∈J) = (fAj (a1, . . . , ani

))j∈J

for all a1j, . . . , anij ∈ Πj∈JAj. If J = ∅, then Π∅ is the trivial algebra withuniverse {∅}. If J = {1, . . . , n}, the direct product can be written as A1× . . .×An. If for all j ∈ J, Aj = A, then we write AJ instead of Πj∈JAj

For each i ∈ I, the ith-projection of the direct product Πj∈JAj is themapping

πi: Πj∈JAj → Aj

which is defined by πi(aj)j∈J = ai. One can see that each ith-projection is anepimorphism.

1.22 Definition An algebra A is a subdirect product of an indexed fam-ily (Aj)j∈J of algebras if

(i) A is a subalgebra of Πj∈JAj

(ii) πi(A) = Ai for all i ∈ I.

An embedding α: A → Πj∈JAj is subdirect if α(A) is a subdirect product of(Aj)j∈J .

1.23 Definition An algebra A is subdirectly irreducible if for every sub-direct embedding ν: A → Πi∈IA/θi, there is an i ∈ I such that π ◦ ν: A → Ai isan isomorphism.

1.24 Example Every simple algebra is subdirectly irreducible.

A major theme in universal algebras is the study of classes of algebrasof the same type closed under one or more the following constructions.

1.25 Definition We introduce the following operators mapping from aclass of algebras to a class of algebras. If K ⊆ Alg(τ), then

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A ∈I(K) ⇐⇒ A is isomorphic to some members of KA ∈H(K) ⇐⇒ A is a homomorphic image of some members of KA ∈S(K) ⇐⇒ A is a subalgebra of some members of KA ∈P(K) ⇐⇒ A is a direct product of a family of algebras of K

1.26 Definition A nonempty class K of algebras of type τ is called avariety if it is closed under subalgebras, homomorphic images and direct prod-ucts.

As the intersection of classes of varieties of type τ is also variety and Alg(τ) isa variety, we can conclude that for every class K of algebras of the same type,there is a smallest variety containing K.

Let K be a class of algebras of the same type and let V (K) denotethe smallest variety containing K. We say that V (K) is the variety generatedby K. If K has a single menber A, then we write simply V (A). A variety V isfinitely generated if V = V (K) for some finite set of finite algebras K.

The following theorem and corollary were proved in [1].

1.27 Theorem [1] Let K be a class of algebras of type τ . Then the classHSP (K) = V (K).

1.28 Theorem [1] Let K be a class of algebras of type τ . ThenK,S(K), H(K), P (K) and V (K) satisfy the same identities.

An algebra A of type τ is called locally finite if every finitely generatedsubalgebra of A is finite. A class K of algebras of the same type is called locallyfinite if every member of K is locally finite.

The following theorem was proved in [5].

1.29 Theorem [5] Let K ⊆ Alg(τ) be a finite set of finite algebras. ThenV (K) is a locally finite.

1.5 Terms and Polynomials

Given an algebra A, there are usually many functions beside the fun-damental operations which are compatible with congruences of A or ”preserve”subalgebras of A. The most obvious functions of this kind are those obtainedby compositions of the fundamental operations. This leads us to the study ofterms.

1.30 Definition Let X be a set of objects called variable. Let τ be atype of algebras. Then the set T (X) of terms of type τ over X is the smallestset such that

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(i) X ⊆ T (X)

(ii) If t1, . . . , tn ∈ T (X) and f is a n-ary operation symbol for somepositive integer n, then f(t1, . . . , tn) ∈ T (X).

For t ∈ T (X), we write t as t(x1, . . . , xn) to indicate that the vari-able occuring in t are x1, . . . , xn. A term t is n-ary if the number of variablesappearing explicitly in t is less than n.

1.31 Example (1) Let τ consist of a single binary operation sym-bol · and let X = {x, y, z}. Then

x, y, z, x · y, y · z, (x · y) · z and ((x · y) · z) · x

are some of the terms over X.

(2) Let τ consist of two binary operations symbol · and + , and letX = {x, y, z}. Then

x, y, z, x · y, y + z, x · (y + z) and x · y + x · z

are some of the terms over X. ¯

Let A be an algebra. We can interpret the variables of terms by ele-ments of the set A and we can interpret the operation symbols by fundamentaloperations of the algebra. This process produces term operations from terms.

1.32 Definition Given a term t(x1, . . . , xn) of type τ over some set Xand given an algebra A of type τ . We define a mapping tA: An → A as follows:

(i) if t is a variable xi for some i, then t(a1, . . . , an) = ai for alla1, . . . , an ∈ A; that is, tA is the i-th projection operation or

(ii) if t is of the form f(t1(x1, . . . , xn), . . . , tk(x1, . . . , xn)) wheref ∈ τ , then

tA(a1, . . . , an) = fA(tA1 (a1, . . . , an), . . . , tAk (a1, . . . , an))

for all a1, . . . , an ∈ A.

Then tA is called a term operation of A. We denote the set of all term operationson A by T (A).

The next thoerem which was proved in [1] gives some useful propertiesof term operations.

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1.33 Theorem [1] For any type τ and algebras A, B of type τ , we havethe followings.

(i) Let t be a n-ary term of type τ and let θ ∈ ConA. Assumethat (ai, bi) ∈ θ for 1 ≤ i ≤ n. Then

tA(a1, . . . , an) θ tA(b1, . . . bn)

(ii) If t is a n-ary term of type τ and α: A → B is a homomorphism,then

α(tA(a1, . . . , an)) = tB(α(a1), . . . , α(an))

for all a1, . . . , an ∈ A.

(iii) If B is a subalgebra of A, then tA(b1, . . . bn) ∈ B for allb1, . . . , bn ∈ B.

Now we will define polynomials. Like terms, polynomials are expres-sions in forms of the inductive compositions of variables and operation symbols.

1.34 Definition Let X be a set of objects called variable. Let τ bea type of algebra. Then the set P (X) of polynomials of type τ over X is thesmallest set such that

(i) X ⊆ P (X)

(ii) if a ∈ A which is the set of all constant symbols, then a ∈ P (X)

(iii) if p1, . . . , pn ∈ P (X) and f is a n-ary operation symbol, thenf(p1, . . . , pn) ∈ P (X).

Like terms, we can interpret the variables of polynomials by elementsof the universe of an algebra and we can interpret an operation symbols byfundamental operations of the algebra.

1.35 Definition Given a term p(x1, . . . , xn) of type τ over some setX and given an algebra A of type τ , we define a mapping pA: An → A as follows:

(i) if p is a variable xi for some i, then p(a1, . . . , an) = ai for alla1, . . . , an ∈ A

(ii) if a ∈ A, then a(x1, . . . , xn) = a for all x1, . . . , xn ∈ A

(iii) if p is of the form f(p1(x1, . . . , xn), . . . , pk(x1, . . . , xn)) wheref ∈ τ , then

p(a1, . . . , an) = f(p1(a1, . . . , an), . . . , pk(a1, . . . , an))

16

for all a1, . . . , an ∈ A.

Then pA is called a polynomial operation of A. We denote the set of all polyno-mial operations on A by P (A).


1.36 Theorem [6] Let A be an algebra. Then

(i) T (A) is the clone generated by all fundamental operations ofA

(ii) P (A) is the clone generated by all fundamental operations ofA together with all constant operations on A.

The following lemma which was proved in [5] characterizes all congru-ence relations of an algebra.

1.37 Lemma [5] Let θ be an equivalence relation on A. The relation θis compatible with all unary polynomial operations of an algebra A if and onlyif θ ∈ ConA.

1.38 Definition Let A = (A; FA1 ) and A′ = (A; FA

2 ) be algebras. ThenA and A′ are called term equivalent if T (A) = T (A′) and A, A′ are calledpolynomially equivalent if P (A) = P (A′).

1.39 Example The two-element algebra ({0, 1};∧,⇒,¬) and ({0, 1};∧,⇒,¬, 0, 1) are term equivalent. Let (G; ·, −1, 1) and (G; ·) are groups.Then they are term equivalent. The two element algebras ({0, 1};∧,∨) and({0, 1};∧,∨, 0, 1) are polynomially equivalent. ¯

1.6 Identities and Mal’cev Conditions

In 1950, A.I. Mal’cev showed the connection between permutability ofcongruence lattices for all algebras in a variety V and the existence of a ternaryterm p such that V satisfies certain identities involving p. The characterizationof properties in varieties by the existence of certain terms involved in certainidentities we will refer to as Mal’cev condition.

1.40 Definition [1] An identity of type τ over X is an expression of theform p ≈ q where p, q ∈ T (X). An algebra A of type τ satisfies an identity p ≈ qwhich is written by A |= p ≈ q if pA = qA. A class of algebras of type τ satisfiesp ≈ q which is written by K |= p ≈ q if A |= p ≈ q for all A ∈ K.

A class K of algebras of type τ is called congruence permutable if everyalgebra A in K is congruence permutable.

17

1.41 Theorem (Mal’cev)[1] Let V be a variety of type τ . Then V iscongruence permutable if and only if there is a term p(x, y, z) such that

V |= p(x, x, y) ≈ y and V |= p(x, y, y) ≈ x.

1.42 Example It is clear that the variety of groups satisfies the fol-lowing identities:

(xy)z ≈ x(yz), ex ≈ x ≈ xe and xx−1 ≈ x−1x ≈ e.

We can define a term p by p(x, y, z) = xy−1z such that every group satisfiesp(x, x, y) ≈ xx−1y ≈ y and p(x, y, y) ≈ xy−1y ≈ x; so, the variety of all groupsis congruence permutable.

Similarly, the variety of all rings is congruence permutable since wecan define a Malcev’ term given by p(x, y, z) = x− y + z for the variety; so, thevariety of rings is also congruence permutable. ¯

A class K of algebras of type τ is called congruence distributive if everyalgebra A in K is congruence distributive. The following theorem which wasproved in [1] gives a Mal’cev type condition which is sufficient for a congruencedistributive.

1.43 Theorem [1] Let V be a variety with a ternary term m such that

V |= m(x, x, y) ≈ m(x, y, x) ≈ p(y, x, x) ≈ x.

Then V is congruence distributive.

1.44 Example Let m be the ternary lattice term defined by

m(x, y, z) = (x ∨ y) ∧ (x ∨ z) ∧ (y ∨ z).

Then for any lattice, we have

m(x, x, y) = (x ∨ x) ∧ (x ∨ y) ∧ (x ∨ y) ≈ x ∧ (x ∨ y) ≈ x

m(x, y, x) = (x ∨ y) ∧ (x ∨ x) ∧ (y ∨ x) ≈ (x ∨ y) ∧ x ≈ x

m(y, x, x) = (y ∨ x) ∧ (y ∨ x) ∧ (x ∨ x) ≈ (y ∨ x) ∧ x ≈ x.

Thus the variety of all lattices is congruence distributive. ¯A class K of algebras of type τ is called congruence modular if every

algebra A in K is congruence modular.

The following lemma was proved in [1].

18

1.45 Lemma [1] If an algebra A is congruence permutable (congruencedistributive), then A is congruence modular.

A class K of algebras of type τ is called n-permutable if every algebraA in K is n-permutable.

The following theorem which was proved by J.A Hagemann and A.Mitschke in [1] gives the Mal’cev type conditions for n-permutable variety.

1.46 Theorem [6] Let n ≥ 2. A variety V is n-permutable if and only ifthere are ternary terms p0(x, y, z), . . . , pn(x, y, z) such that V satisfies

p0(x, y, z) ≈ x

pi(x, x, y) ≈ pi+1(x, y, y) for all 0 ≤ i ≥ n

pi(x, y, x) ≈ x

1.47 Lemma [6] If a two-elements algebra A is n-permutable, then A iseither 2-permutable or 3-permutable.

Chapter 2

Minimal Algebras

Every algebra which we dealt with in this and following chapters isfinite even when we omit the word ”finite”.

At the beginning of the eighties, R. McKenzie and D. Hobby developeda new theory which is called ”Tame Congruence Theory ”. An important conceptin this theory is the concept of a minimal algebra.

In this chapter, we will investigate an induced algebra of an algebrawhich is a tool to define a minimal algebra and we study some important prop-erties of a minimal algebra which inherits from the belonging algebra.

Let A and B be sets, let f : A → B be a mapping and let U be asubset of A. Then the restriction of f to the set U is the mapping f|U from Uto B defined by f|U (x) := f(x) for all x ∈ U . We will be particularly interestedin the restrictions of polynomial operations on an algebra A to certain specialsubsets of the base set A. To describe a minimal algebra, we need the followingdefinition.

2.1 Definition [5] Let A be an algebra and let U be a subset of the universeA. We define the following restrictions to U .

(i) If θ is an equivalence relation on A, we define θ|U := θ ∩ U2.

(ii) If g is a n-ary operation on A, we define g|U := g|Un

(iii) We define P (A)|U to be the set of g|U where g is a n-ary oper-ation of A and g(Un) ⊆ U for some n ∈ N.

(iv) We define A|U to be the algebra (U ; P (A|U )).

It is not difficult to show that θ∩U2 is an equivalence relation on U . The relationθ|U is called the restriction of θ . An operation g|U is also called the restrictionof g to U . The algebra A|U is called the induced algebra of A on U .

We will first prove that P (A)|U = P (A|U ).

19

20

2.2 Proposition Every polynomial operation of A|U is a fundamentaloperation of A|U .

Proof: Let A be an algebra.

(i) We show that all constant operations on U are contained inP (A)|U . Let cu ∈ {cu |u ∈ U}. Then there is a c′u ∈ P (A) such that c′u(x) = ufor all x ∈ A and c′u|U = cu. So, cu ∈ P (A)|U .

(ii) We show that P (A)|U is a clone. Let g be a k-ary fundamentaloperation of A|U and h1, ..., hk be n-ary fundamental operations of A|U . Bythe definition of P (A)|U , there are a k-ary polynomial operation g′ and n-arypolynomial operations h′1, ..., h

′k such that g = g′|U , g′(Uk) ⊆ U , hi = h′i|U and

h′i(Un) ⊆ U for all 1 ≤ i ≤ n. It is clear that g′(h′1, ..., h

′k)(U

n) ⊆ U . Leta1, ..., an ∈ U . Then

g(h1, ..., hk)(a1, ..., an) = g(h1(a1, ..., an), ..., hk(a1, ..., an))

= g′|U (h′1|U (a1, ..., an), ..., h′k |U (a1, ..., an))

= g′|U (h′1(a1, ..., an), ..., h′k(a1, ..., an))

= g′(h′1(a1, ..., an), ..., h′k(a1, ..., an))

= g′(h′1, ..., h′k)(a1, ..., an)

= g′(h′1, ..., h′k)|U (a1, ..., an) ∈ U

Since P (A) is a clone, g′(h′1, ..., h′k) ∈ P (A). So, g(h1, ..., hk) ∈ P (A)|U . Since

en,Ai ∈ P (A) for all i and for all n, we have en,A

i |U ∈ P (A)|U ; so, P (A)|U containsall projection operations on U

(iii) Obviously, P (A)|U ⊆ P (A|U ) = < P (A)|U ∪ {cu |u ∈ U} >.Since {cu | u ∈ U} ⊆ P (A)|U , we have P (A|U ) = < P (A)|U >. Since P (A|U ) isthe smallest clone containinig P (A)|U , we have P (A|U ) = P (A)|U . 2

To obtain a minimal algebra, we consider an image of a mapping;particularly, those of idempotent polynomial mappings on the algebra.

2.3 Definition [5] A mapping e : A → A is called idempotent if e2 = e◦e =e. For an algebra A , we denote the set of all idempotent polynomial operationsof A by E(A).

We have the following proposition which is useful for the proof in se-quel.

2.4 Proposition Let e be an idempotent mapping on A and U = e(A).Then e|U is an identity mapping on U .

Proof: Let b ∈ U . Then there is an element a ∈ A such that e(a) = bwhich implies that e(e(a)) = e(b); since e2 = e, we have e(b) = e(e(a)) = e(a) =b. 2

21

The following theorem, which was proved in [5], gives some sufficientconditions for the restriction of a congruence on A to a subset U of A to be alsoa congruence relation on the induced algebra A|U .

2.5 Theorem [5] Let A be an algebra and let e ∈ E(A). If U = e(A),then the mapping ϕ: Con(A) → Con(A|U ), which is defined by ϕ(θ) = θ|U , is alattice epimorphism.

The following corollary follow from Homomorphism Theorem[1].

2.6 Corollary If U satisfies the assumption of Theorem 2.5, thenCon(A)/kerϕ ∼= Con(A|U ).

We are now ready to define a minimal set of an algebra A. It will bean image of A under a unary operation.

2.7 Definition [5] Let A be an algebra and let f be a unary polynomialoperation of A. The image set f(A) is called a minimal set of A if the followingconditions are satisfied;

(i) |f(A)| > 1

(ii) if g is a unary polynomial of A where g(A) ⊆ f(A) and |g(A)| >1 then g(A) = f(A).

The collection of all minimal sets of an algebra A will be denoted by Min(A).An induced algebra A|U , which is induced by a minimal set U ∈ Min(A), iscalled a minimal algebra of A. We have the following proposition.

2.8 Proposition Let A be a nontrivial algebra. If P (A) contains a non-constant unary polynomial operation, then Min(A) 6= ∅.

Proof: Let h be a non-constant unary polynomial operation of A. Then|h(A)| ≥ 2. Clearly, h(A) is a minimal set of A if |h(A)| = 2. Assume that|h(A)| > 2. Then L = {|g(A)| | g is a non-constant unary polynomial opera-tion of A} has the least element. Denote the least element of L by l. So, thereis a non-constant unary polynomial f such that |f(A)| = l > 1.

To show that f(A) is a minimal set of A, let g be a unary polynomialof A such that g(A) ⊆ f(A) and |g(A)| > 1. Then |g(A)| ∈ L; hence, |g(A)| ≥|f(A)|; so, |g(A)| = |f(A)|. Since g(A) ⊆ f(A) and A is finite, we have g(A) =f(A). Thus, f(A) ∈ Min(A). 2

2.9 Example Let consider the algebra

A = ({0, 1, 2}; max(x, y), min(x, y), 0, 1, 2 m1,m2)

22

where m1 and m2 are unary operations defined as in Table 2.1

x m1 m2

0 0 01 2 02 2 2.

Table 2.1

Let f1, . . . , f6 be operations on A defined as in Table 2.2

x f1 f2 f3 f4 f5 f6

0 0 0 0 0 1 11 0 1 0 2 1 22 1 1 2 2 2 2.

Table 2.2

It is clear that f1, . . . , f6 are unary polynomial operations of A which are ex-amples of unary polynomial operations such that the corresponding image setsfi({0, 1, 2}) are minimal sets for all i = 1, . . . , 6. ¯


2.10 Theorem [5] Let A be an algebra and let U ∈ Min(A). Then everyunary polynomial operation of the minimal algebra A|U is either a permutationor a constant operation.

The images of idempotent polynomial mappings on an algebra play animportant role. So, we need some sufficient conditions for a minimal set of analgebra to be an image of an idempotent polynomial operation of the algebra.To describe the conditions involved, we need the following definition.

2.11 Definition [5] Let L = (L,∧,∨) be a lattice.

(i) A mapping µ : L → L is called a meet-endomorphism of L if itsatisfies µ(x ∧ y) = µ(x) ∧ µ(y) for all elements x and y in L.

(ii) A mapping µ : L → L is called a join-endomorphism of L if itsatisfies µ(x ∨ y) = µ(x) ∨ µ(y) for all elements x and y in L.

(iii) A mapping µ : L → L is called extensive if it satisfies x ≤ µ(x)for all elements x in L; and a mapping µ is strongly extensive if x < µ(x) for allelements x in L which are not the greatest element of L.

23

The following theorem, which was proved in [5], shows the conditionwhich implies that every minimal set is an image of an idempotent operation ofan algebra.

2.12 Theorem [5] Let A be an algebra. Assume that Con(A) has nonon-constant strongly extensive meet endomorphism. Then every minimal setU of A has the form U = e(A) for some e ∈ E(A).

2.13 Lemma [5] Let L be a finite lattice with the least element 0, thegreatest element 1 and the property that the meet of arbitrary coatoms is zero.Then L has no non-constant strongly extensive meet endomorphism.

Lemma 2.13 can be applied to the lattices Mn of the form

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@@

@

HHHHHH

³³³³³³³³³

PPPPPPPPP

@@

@

¡¡

¡

©©©©©©

PPPPPPPPP

³³³³³³³³³

t

tttt tr r r

t

Figure 1

The following definition introduces those properties of the congruencelattice of an algebra A which are important for tame algebras.

2.14 Definition [5] Let L be a lattice with the least element 0 and thegreatest element 1. A homomorphism ϕ : L → L′ is called 0-separating if

ϕ−1(ϕ(0)) = {0},

and ϕ is called 1-separating if

ϕ−1(ϕ(1)) = {1}.

A homomorphism ϕ is called 0-1-separating if ϕ is both 0-separating and 1-separating. A lattice L is called 0-1-simple if |L| > 1 and every non-constanthomomorphism f is 0-1-separating. A nontrivial finite lattice L is called tightif L is 0-1-simple and L has no non-constant strongly extensive meet endomor-phism.

24

It is well-known that there is a 1-1 correspondence between the setof all congruence relations on an algebra and the set of all homomorphisms ofthe algebra. It is natural to claim that the congruence class containing 0 withrespect to kerϕ is singleton if the homomorphism ϕ of a lattice L with 0 and 1is 0-separating and similarly for 1-separating.

We will prove the following proposition

2.15 Proposition Let A be an algebra and U = e(A) for some e ∈ E(A).Then Con(A|U ) satisfies the following properties if Con(A) also does

(i) 0-1-simple

(ii) no non-constant meet-endomorphism

(iii) no non-constant strongly extensive

(iv) no non-constant strongly extensive meet-endomorphism

(v) tight.

Proof: Let A be a finite algebra and U = e(A) for some e ∈ E(A).

(i) Let L be a lattice and let f : Con(A|U ) → L be a non-constanthomomorphism. By Theorem 2.5, there is an epimorphism ϕ : Con(A) →Con(A|U ) which is defined by ϕ(θ) = θ|U . So, fϕ is a non-constant homo-morphism from Con(A) to L which implies that fϕ is 0-1 separating. Hence,{∆A} = (fϕ)−1(fϕ)(∆A) = ϕ−1(f−1(f(ϕ(∆A)))) = ϕ−1(f−1(f(∆U))) and{A×A} = (fϕ)−1(fϕ)(A×A) = ϕ−1(f−1(f(ϕ(A×A)))) = ϕ−1(f−1(f(U×U))).Therefore, ϕ(ϕ−1(f−1(f(∆U)))) = ϕ({∆A}) = {∆U} and ϕ(ϕ−1(f−1(f(U ×U)))) = ϕ({A × A}) = {U × U}. Since ϕ is surjective, f−1(f(∆U)) = {∆U}and f−1(f(U × U)) = {U × U} which imply that f is 0-1-separating. Thus,Con(A|U ) is 0-1-simple.

(ii) Suppose that f is a non-constant meet-endomorphism ofCon(A|U ). Then f(∆U) ⊂ f(U × U) which implies that f(U × U) 6= ∆U .We define g : Con(A|U ) → Con(A) for each θ ∈ Con(A|U ) by

g(θ) =

{A× A, f(U × U) ⊆ θ∆A, otherwise.

We show that g is a meet-homomorphism. Let θ, β ∈ Con(A|U ). If f(U × U) ⊆θ∩β then f(U×U) ⊆ θ and f(U×U) ⊆ β which imply that g(θ) = A×A = g(β).So, g(θ) ∩ g(β) = A×A = g(θ ∩ β). If f(U × U) 6⊆ θ ∩ β then f(U × U) 6⊆ θ orf(U×U) 6⊆ β which imply that g(θ) = ∆A or g(β) = ∆A. So, g(θ)∩g(β) = ∆A =g(θ∩β). Thus, g is a meet-homomorphism. Since ϕ is a lattice homomorphism,ϕ is a meet-homomorphism. So, gϕ is a meet-endomorphism of Con(A) whichimplies that gϕ is constant. Since ∆U ⊂ f(U × U) and f(U × U) ⊆ U × U , weget gϕ(∆A) = g(∆U) = ∆A and gϕ(A × A) = g(U × U) = A × A which imply

25

that gϕ is non-constant; a contradiction. Thus, Con(A|U ) has no non-constantmeet-endomorphism.

(iii) Suppose that h is a non-constant strongly extensive ofCon(A|U ). Then h(U × U) = U × U and there is a congruence φ ∈ Con(A|U )such that h(φ) ⊂ h(U × U). Since U = e(A) and Theorem 2.5 imply that thereis a congruence φ ∈ Con(A) such that φ|U = φ; so, h(φ|U ) = h(φ) ⊂ h(U × U).We define µ: Con(A|U ) → Con(A) for each θ ∈ Con(A|U ) by

µ(θ) = θ = {(x, y) ∈ A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ θ}

where U(A) is the set of all unary polynomial operations of A.

(a) We show that θ ∈ Con(A) and θ|U = θ. It is clearthat θ is an equivalence relation on A. For the congruence property, we showthat θ is compatible with all unary polynomial operations of A. Let g be aunary polynomial operation of A and (a, b) ∈ θ. Then for each unary polynomialoperation f , the composition operation f◦g is a unary polynomial operation of A.So, the definition of θ implies (e(f◦g)(a), e(f◦g)(b)) = (e(f(g(a))), e(f(g(b)))) ∈θ for all f ∈ U(A) which imply that (g(a), g(b)) ∈ θ. To show that θ|U = θ,let (a, b) ∈ θ|U . Then (a, b) ∈ θ and (a, b) ∈ U2. Since e is a unary polynomialopeation of A, we have (e(e(a)), e(e(b))) ∈ θ. Since e|U is an identity operationon U , we have (a, b) ∈ θ. Assume that (a, b) ∈ θ. Then (a, b) ∈ U × U andlet f ∈ U(A). Then e(f(U)) ⊆ U which implies that (e ◦ f)|U is a fundamentaloperation of A|U . So, (e(f(a)), e(f(b))) ∈ θ for all f ∈ U(A) which imply that(a, b) ∈ θ. Thus, (a, b) ∈ θ|U .

(b) We show that β ⊆ θ if and only if β|U ⊆ θ. Assumethat (a, b) ∈ β|U . Then (a, b) ∈ β ⊆ θ and (a, b) ∈ U2 which imply that(a, b) ∈ θ|U = θ. Conversely, we assume that β 6⊆ θ. Then, there is a (x, y) ∈ βsuch that (x, y) 6∈ θ. So, there is a unary polynomial operation h such that(e(h(x)), e(h(y))) 6∈ θ. But (e(h(x)), e(h(y))) ∈ β|U . So, β|U 6⊆ θ.

(c) We show that if β|U ⊂ θ then β ⊂ θ. By using (b), wehave β ⊆ θ. Since β|U ⊂ θ, there exist (a, b) ∈ θ such that (a, b) 6∈ β|U whichimplies that (a, b) ∈ θ but (a, b) 6∈ β. Thus, β ⊂ θ.

(d) We show that µhϕ is a strongly extensive mapping ofCon(A). Let θ′ ∈ Con(A). If θ′ = A × A then µhϕ(A × A) = µh(U × U) =µ(U × U) = A × A. If θ′ 6= A × A then µhϕ(θ′) = µ(h(θ′|U )) = {(x, y) ∈A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ h(θ′|U )} = β. Since h is strongly ex-

tensive, θ′|U ⊂ h(θ′|U ) = β|U . By (c), θ′ ⊂ β = µhϕ(θ′); hence, µhϕ is astrongly extensive mapping of Con(A) which implies that µhϕ is constant.Since φ and A × A ∈ Con(A), we have µ(h(ϕ(φ))) = µ(h(φ)) = {(x, y) ∈A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ h(φ)} = θ and µ(h(ϕ(A × A))) =µ(h(U × U)) = {(x, y) ∈ A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ h(U × U)} = β.So, θ|U = h(φ) and β|U = h(U × U). Since h(φ) ⊂ h(U × U), we have

26

θ|U = h(φ) ⊂ h(U × U) = β|U . By (c), we have θ ⊂ β. So, µhϕ is non-constantwhich is a contradiction.

(iv) Suppose that h is a strongly extensive meet-endomorphism ofCon(A|U ). By using a mapping µ in (iii), µhϕ is strongly extensive mapping ofCon(A). Next, we show that µ is a meet-homomorphism. Let θ, β ∈ Con(A|U ).Then

µ(θ ∩ β) = {(x, y) ∈ A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ θ ∩ β},µ(β) = {(x, y) ∈ A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ θ} and

µ(θ) = {(x, y) ∈ A2 | ∀f ∈ U(A), (e(f(x)), e(f(y))) ∈ θ}.To show that µ(θ ∩ β) = µ(θ) ∩ µ(β), let (x, y) ∈ µ(θ ∩ β). Then(e(f(x)), e(f(y))) ∈ θ ∩ β for all f ∈ U(A); so, (e(f(x)), e(f(y))) ∈ θ and(e(f(x)), e(f(y))) ∈ β for all f ∈ U(A) which imply that (x, y) ∈ µ(θ) ∩ µ(β).By a similar argument, µ(θ)∩µ(β) ⊆ µ(θ∩β). So, µhϕ is a meet-endomorphismof Con(A). Thus, µhϕ is a strongly extensive meet-endomorphism of Con(A)which implies that µhϕ is constant. But we have proved in (iii) that there is aθ ∈ Con(A) such that µhϕ(θ) 6= µhϕ(A× A) which is a contradiction.

(v) is a consequence of (i) and (iv). 2

Chapter 3

Some Properties of TameAlgebras

Let A be an algebra. Recall that a minimal algebra A|U is an inducedalgebra in which U is a minimal set of A.

K. Denecke[5] introduced concept of tame in Tame Congruence Theoryand gave some sufficient condition for algebras to be tame. In this chapter, wewill give some sufficient conditions for a minimal algebra to be tame. Moreover,we will study some properties of minimal algebras of a tame algebra.

3.1 Definition [5] A finite algebra A is called tame if it has a minimal setU ∈ Min(A) satisfied the following conditions

(i) there is an e ∈ E(A) with U = e(A)

(ii) if ∆A ⊂ θ then ∆U ⊂ θ|U

(iii) if θ ⊂ A× A then θ|U ⊂ U × U .

3.2 Definition [5] Let A be an algebra and let B and C be subsets of theuniverse set A of A. Then the sets B and C are called polynomially isomorphicin A if there are unary polynomials f and g of A such that

f(B) = C, g(C) = B, gf |B = idB and fg|C = idC .

In this case the mapping f|B is called a polynomially isomorphism from B ontoC.

The following lemma and theorem were proved in [5].

3.3 Lemma [5] Let B and C be polynomially isomorphic subsets in A.Then f|B : B → C is an isomorphism from A|B onto A|C .

27

28

3.4 Theorem [5] Let A be a tame algebra. Then

(i) for all U1, U2 ∈ Min(A), the minimal algebras A|U1and A|U2

are polynomially isomorphic in A

(ii) for each U ∈ Min(A), U satisfies Conditions (i),(ii),(iii) ofDefinition 3.1.

(iii) there is a unary polynomial f of A such that |f(A)| > 1 andf(U) ∈ Min(A) and f|U is a polynomial isomorphism from U onto f(U).

We have the following proposition.

3.5 Proposition Let A be a tame algebra and f be a unary polynomialoperation of A. Then |f(A)| = |U | for some U ∈ Min(A) if and only if f(A) isa minimal set of A.

Proof: Let f be a unary polynomial operation of A and U ∈ Min(A)such that |f(A)| = |U |. Then |f(A)| > 1. To show that f(A) ∈ Min(A), letgA be a unary polynomial of A such that g(A) ⊆ f(A) and |g(A)| > 1. Wedefine L = {|h(A)| |h ∈ U(A), h(A) ⊆ f(A) and |h(A)| > 1} where U(A) is theset of all unary polynomial operations of A. Then L contains the least element.Denote the least element of L by l. So, there is a unary polynomial h of A suchthat h(A) ⊆ f(A) and |h(A)| = l > 1. To show that h(A) ∈ Min(A), let p bea unary polynomial of A such that p(A) ⊆ h(A) ⊆ f(A) and |p(A)| > 1. Then|p(A)| ∈ L which implies that |p(A)| ≥ |h(A)|; since p(A) ⊆ h(A), we havep(A) = h(A). Thus, h(A) ∈ Min(A). Since A is tame and Theorem 3.4 impliesthat l = |h(A)| = |U |; so, |h(A)| = |f(A)|. Since |g(A)| ∈ L and g(A) ⊆ f(A),we have |U | ≤ |g(A)| ≤ |f(A)| = |U |; so, |g(A)| = |U | = |f(A)| which impliesthat g(A) = f(A). Thus, f(A) ∈ Min(A). Conversly, the result follows fromLemma 3.3 and Theorem 3.4. 2

Let f : A → B be a function and let θ be a binary relation on A. Wedefine f(θ) := {(f(a), f(b)) | (a, b) ∈ θ}. The relation f(θ) is called an image ofθ.

We know that the image of a congruence of an algebra A under a sur-jective homomorphism needs not to be congruence. In the following proposition,we will give a sufficient condition for the image of a congruence of A under asurjective homomorphism to be also a congruence relation.

3.6 Proposition Let ϕ : A → B be a surjective homomorphism. Then

(i) ϕ(β) ∈ Con(B) for all β ∈ [kerϕ, A× A],

(ii) there is a lattice isomorphism h from [kerϕ, A×A] to Con(B).

29

Proof: Let ϕ: A → B be a surjective homomorphism.

(i) We show that ϕ(β) is a congruence relation on B. Clearly,the relation ϕ(β) is reflexive and symmetric. To show that ϕ(β) is transitive,let (a, b), (b, c) ∈ ϕ(β). Then there are elements (x, y), (u, v) ∈ β such that(ϕ(x), ϕ(y)) = (a, b) and (ϕ(u), ϕ(v)) = (b, c) which imply that ϕ(y) = b = ϕ(u).So, (y, u) ∈ kerϕ ⊆ β. By the transitivity of kerϕ, we get (x, v) ∈ kerϕ ⊆ βwhich implies that (a, c) ∈ ϕ(β). Thus, ϕ(β) is an equivalence relation on A.For the congruence property, we show that ϕ(β) is compatible with all unarypolynomial operations of B. Let fB be a unary polynomial operation of B andlet (a, b) ∈ ϕ(β). Then there exists an element (x, y) ∈ β such that ϕ(x) = aand ϕ(y) = b. Since ϕ is a homomorphism, there is a unary polynomial fA of Acorresponding to fB such that (fA(x), fA(y)) ∈ β and (ϕ(fA(x)), ϕ(fA(y))) ∈ϕ(β). But, ϕ(fA(x)) = fB(ϕ(x)) = fB(a) and ϕ(fA(y)) = fB(ϕ(y)) = fB(b)imply that (fB(a), fB(b)) ∈ ϕ(β).

(ii) We define µ: [kerϕ, A×A] → Con(B) for each θ ∈ [kerϕ,A×A] by µ(θ) = ϕ(θ).

(a) First, we show that µ is an order-embedding mapping;that is, θ ⊆ β if and only if µ(θ) ⊆ µ(β) for all θ, β ∈ Con(A). Assume that(x, y) ∈ µ(θ). Then there is an element (a, b) ∈ θ such that (ϕ(a), ϕ(b)) = (x, y).Since θ ⊆ β, we have (x, y) ∈ µ(β). Conversely, if (a, b) ∈ θ then (ϕ(a), ϕ(b)) ∈µ(θ) ⊆ µ(β); and so, there is an element (c, d) ∈ β such that ϕ(c) = ϕ(a) andϕ(d) = ϕ(b) which imply that (a, c), (b, d) ∈ kerϕ ⊆ β; hence, (a, b) ∈ β.

(b) Now, we show that µ is surjective. Let θ be a congru-ence relation of B. We define the relation

θ: = {(x, y) ∈ A× A|(ϕ(x), ϕ(y)) ∈ θ}

and we prove that θ is a congruence on A whose image under µ is θ. Obviously,θ is an equivalence relation on A. To show the congruence property of θ, let fA

be a unary polynomial operation of A and let (x, y) ∈ θ. Then (ϕ(x), ϕ(y)) ∈ θ;hence, there is a unary polynomial operation fB of B such that fB correspondsto fA and (fB(ϕ(x)), fB(ϕ(y))) ∈ θ. But fB(ϕ(x)) = ϕ(fA(x)) and fB(ϕ(y)) =ϕ(fA(y)), we have (ϕ(fA(x)), ϕ(fA(y))) ∈ θ which implies that (fA(x), fA(y)) ∈θ. To show that ϕ(θ) = θ, let (a, b) ∈ ϕ(θ). Then there is an element (x, y) ∈ θsuch that (ϕ(x), ϕ(y)) = (a, b). By the definition of θ, we have (a, b) ∈ θ whichimplies that ϕ(θ) ⊆ θ. Conversely, if (a, b) ∈ θ then (a, b) ∈ B × B whichimplies that there are elements x, y ∈ A such that ϕ(x) = a and ϕ(y) = b; so,(x, y) ∈ θ which implies that (a, b) = (ϕ(x), ϕ(y)) ∈ ϕ(θ). Thus, µ is a surjectiveorder-embedding mapping which implies that µ is also a lattice-homomorphism.2

Let h : A → B be a homomorphism from A into B. It is well-knownthat h(A) is a subalgebra of B. If A is tame, we want that h(A) is tame.

30

Obviously, if h is constant, then h(A) is not tame. So, we look for the propertiesof a homomorphism which can tell us that the homomorphic image of a tamealgebra is tame. Now, we give a necessary and sufficient condition for thosehomomorphisms.

3.7 Theorem Let ϕ : A → B be a non-constant homomorphism froma tame algebra A into an algebra B. Then ϕ satisfies Conditions (a) and (b) forsome U ∈ Min(A):

(a) ϕ(A− U) ∩ ϕ(U) = ∅, and

(b) kerϕ ⊂ θ ⇒ kerϕ|U ⊂ θ|U for all kerϕ 6= θ ∈ Con(A)

if and only if ϕ satisfies the following Condition (i) and (ii):

(i) the homomorphic image of A under ϕ is a tame subalgebra ofB, and

(ii) there is a minimal set V of ϕ(A) such that ϕ−1(V ) is a minimalset of A.

Proof: Let ϕ: A → B be a non-constant homomorphism from a tame algebraA into an algebra B. Then ϕ(A) is a nontrivial subalgebra of B.

To show that ϕ(A) is tame, let U ∈ Min(A) satisfy Conditions (a) and(b). Since ϕ is non-constant , kerϕ ⊂ A×A which implies that kerϕ|U ⊂ U×U .So, there are elements a, b ∈ U such that ϕ(a) 6= ϕ(b); hence, |ϕ(U)| > 1. Toshow that ϕ−1(ϕ(U)) = U , we suppose that U ⊂ ϕ−1(ϕ(U)). Then there is anelement a ∈ ϕ−1(ϕ(U)) such that a 6∈ U . So, there is an element b ∈ ϕ(U) suchthat ϕ(a) = b which contradicts to Condition (a).

To show that ϕ(U) ∈ Min(ϕ(A)), let gϕ(A) be a non-constant unarypolynomial operation of ϕ(A) such that gϕ(A)(ϕ(A)) ⊆ ϕ(U). Then, there isa unary polynomial gA of A corresponding to gϕ(A) such that |gA(A)| > 1 andgA(A) ⊆ ϕ−1(ϕ(gA(A))) = ϕ−1(gϕ(A)(ϕ(A))) ⊆ ϕ−1(ϕ(U)) = U . By the min-imality of U , we have gA(A) = fA(A); and hence, gϕ(A)(ϕ(A)) = ϕ(gA(A)) =ϕ(U). Therefore, ϕ(U) is a minimal set of ϕ(A).

Next we show that ϕ(U) satisfies Conditions (i)−(iii) of Definition 3.1.Let θ ∈ Con(ϕ(A)). Assume that ∆ϕ(A) ⊂ θ. Then there is a θ ∈ [kerϕ, A× A]such that ϕ(kerϕ) = ∆ϕ(A) ⊂ θ = ϕ(θ). So, kerϕ ⊂ θ which implies thatkerϕ|U ⊂ θ|U . Clearly, ϕ(kerϕ|U ) ⊆ ϕ(θ|U ). To prove that ϕ(kerϕ|U ) ⊂ ϕ(θ|U ),

we suppose that ϕ(kerϕ|U ) = ϕ(θ|U ). Since kerϕ|U ⊂ θ|U , there is an element

(a, b) ∈ θ|U such that (a, b) 6∈ kerϕ|U . So, (ϕ(a), ϕ(b)) ∈ ϕ(θ|U ) = ϕ(kerϕ)|ϕ(U)

which implies that there is an element (c, d) ∈ kerϕ|U such that ϕ(c) = ϕ(a)and ϕ(d) = ϕ(b). So, (a, c), (b, d) ∈ kerϕ|U . By the transitivity of kerϕ, wehave (a, b) ∈ kerϕ|U which is a contradiction. Since ϕ(β|U ) = ϕ(β) ∩ ϕ(U)2 for

all β ∈ [kerϕ,A × A], we have ∆ϕ(U) = ϕ(kerϕ)|ϕ(U)= ϕ(kerϕ|U ) ⊂ ϕ(θ|U ) =

31

ϕ(θ)|ϕ(U) = θ|ϕ(U). Thus, ϕ(U) satisfies Condition (ii) of Definition 3.1. By a

similar argument, ϕ(U) satisfies Condition (iii) of Definition 3.1.

Finally we show that ϕ(U) is the image of an idempotent unary poly-nomial operation of ϕ(A). Since A is tame, there is an idempotent polynomialeA of A such that U = eA(A) and there is a unary polynomial eϕ(A) such thateϕ(A) coressponds to eA. Hence, eϕ(A)(ϕ(A)) = ϕ(eA(A)) = ϕ(U). To provethat eϕ(A) ∈ E(ϕ(A)), let b ∈ ϕ(A). Then there is an element a ∈ A such thatϕ(a) = b. Therefore,

eϕ(A)(eϕ(A)(b)) = eϕ(A)(eϕ(A)(ϕ(a)))

= ϕ(eA(eA(a)))

= ϕ(eA(a))

= eϕ(A)(b).

It is clear that ϕ(U) is a minimal set of ϕ(A) with ϕ−1(ϕ(U)) ∈ Min(A). So,Condition (ii) is satisfied.

Conversely, let V be a minimal set of ϕ(A) such that ϕ−1(V ) ∈Min(A).

We show that ϕ−1(V ) satisfies Conditon (a). We suppose that ϕ(A−ϕ−1(V )) ∩ ϕ(ϕ−1(V )) 6= ∅ and let b ∈ ϕ(A − ϕ−1(V )) ∩ ϕ(ϕ−1(V )). Then b ∈ϕ(A−ϕ−1(V )) and b ∈ ϕ(ϕ−1(V )) = V . So, there are elements a ∈ A−ϕ−1(V )and c ∈ ϕ−1(V ) such that ϕ(a) = b = ϕ(c). Hence, a ∈ ϕ−1(V ) which is acontradiction.

Now, let θ ∈ Con(A) with kerϕ ⊂ θ. Then, ∆ϕ(A) = ϕ(kerϕ) ⊂ϕ(θ). Since ϕ(A) is tame, ∆V ⊂ ϕ(θ)|V . So, there is an element (a, b) ∈ϕ(θ)|V = ϕ(θ) ∩ (V × V ) such that (a, b) 6∈ ∆V which implies that there is a(x, y) ∈ θ ∩ (ϕ−1(V ))2 = θ|ϕ−1(V )

such that ϕ(x) = a 6= b = ϕ(y). Therefore,

(x, y) 6∈ ker(ϕ)|ϕ−1(V ). Thus, ker(ϕ)|ϕ−1(V )

⊂ θ|ϕ−1(V ). Therefore, Condition (b)

is satisfied. 2

3.8 Corollary Let ϕ: A → B be a non-constant homomorphism froma tame algebra A into an algebra B. If Min(A) contains an element satisfyingConditons (a) and (b) of Theorem 3.7, then every element of Min(ϕ(A)) is ahomomorphic image of every element of Min(A).

Proof: Let U be a minimal set of A which satisfies Conditons (a) and (b)of Theorem 3.7. Then ϕ(A) is tame and ϕ(U) is a minimal set of ϕ(A). Weshow that ϕ|U : U → ϕ(U) is an epimorphism from A|U onto ϕ(A)|ϕ(U)

. It is clearthat ϕ|U is surjective. Let h be a n-ary fundamental operation of A|U and leta1, ..., an ∈ U . Then, there is a unary polynomial hA of A such that h = hA|U

32

and hA(Un) ⊆ U . So,

ϕ|U (h(a1, ..., an)) = ϕ|U (hA|U (a1, ..., an))

= ϕ|U (hA(a1, ..., an))

= hB(ϕ(a1), ..., ϕ(an))

= hB|ϕ(A)

(ϕ(a1), ..., ϕ(an))

= (hB|ϕ(A)

)|ϕ(U)(ϕ(a1), ..., ϕ(an)).

where hB is the polynomial operation of B corresponding to hA. It is clear thathB |ϕ(A)

is a polynomial operation of ϕ(A). Denote hB |ϕ(A)by hϕ(A). To prove

that hϕ(A)|ϕ(U)∈ P (ϕ(A)|ϕ(U)

), let b1, ..., bn ∈ ϕ(U). Then there are elementsa1, ..., an ∈ U such that b = ϕ(a1), ..., bn = ϕ(an). So,

hϕ(A)(b1, ..., bn) = hϕ(A)(ϕ(a1), ..., ϕ(an))

= ϕ(hA(a1, ..., an)) ∈ ϕ(U).

Hence, hϕ(A)(ϕ(Un)) ⊆ ϕ(U) which implies that hϕ(A)|ϕ(U)∈ P (ϕ(A)|ϕ(U)

). Thus,ϕ|U is a homomorphism. Since A and ϕ(A) are tame, Lemma 3.3 and Theorem3.4 imply that A|U

∼= A|V for all U, V ∈ Min(A) and ϕ(A)|S ∼= ϕ(A)|T forall S, T ∈ Min(ϕ(A)). So, there is an epimorphism from V to S for eachV ∈ Min(A) and S ∈ Min(ϕ(A)). 2

Isomorphism is a special case of homomorphism. The following propo-sition characterizes all isomorphic images of a tame algebra.

3.9 Proposition Let A and B are isomorphic algebras and U ⊆ A. Then

(i) A is tame if and only if B is tame.

(ii) If ϕ is an isomorphism, then U ∈ Min(A) if and only if ϕ(U)is a minimal set of B.

Moreover, if M and N are minimal algebras of A and B, respectivelythen M ∼= N .

Proof: Let ϕ: A → B be an isomorphism.

(i) Assume that A is tame. Then there is a minimal set U of Asuch that U satisfies Conditions (i)-(iii) of Definition 3.1 and Condition (a) ofTheorem 3.7 follows from the injectivity of ϕ. Since kerϕ = ∆A and A is tame,kerϕ|U = ∆A|U

⊂ θ|U for all kerϕ 6= θ ∈ Con(A). Thus, Condition (b) of

Theorem 3.7 is satisfied. Since ϕ is surjective, ϕ(A) = B. Therefore, B is tame.

We can prove the converse statement similarly by using the isomor-phism ϕ−1

(ii) Let U be a minimal set of A. We show that ϕ(U) is a minimalset of B. Since |U | > 1 and ϕ is bijective, |ϕ(U)| > 1. Let gB be a non-constant

33

unary polynomial operation of B such that gB(B) ⊆ ϕ(U). By using the cor-responding gA of A to gB, we have |gA(A)| > 1 and gA(A) = gA(ϕ−1(B)) =ϕ−1(gB(B)) ⊆ ϕ−1(ϕ(U)) = U . By the minimality of U , we have gA(A) = Uwhich implies that gB(B) = ϕ(gA(A)) = ϕ(U).

Finally, we show that ϕ(U) is an image of a unary polynomial operationof B. Since U ∈ Min(A), there is an unary polynomial fA of A such thatfA(A) = U. So, ϕ(U) = ϕ(fA(A)) = fB(B).

We can argue the converse similarly. 2

The following theorem was proved by K. Denecke in [5].

3.10 Theorem [5] Let A be a algebra. If Con(A) is tight, then A is tame.

We have the following propositions.

3.11 Proposition Mn is tight for all n ≥ 3.

Proof: Let n ≥ 3. Then Lemma 2.13 implies that Mn has no non-constant strongly extensive meet-endomorphism. To show that Mn is 0-1-simple,let h: Mn → L be a non-constant lattice homomorphism from Mn into a latticeL. Suppose that h−1(h(0)) 6= {0}. Then there is an element 0 6= a ∈ Mn suchthat h(a) = h(0) which implies that (0, a) ∈ kerh. If a = 1, then h(x) = h(0)for all x ∈ Mn; so, h is constant which is a contradiction. If a 6= 1, then a isan atom(coatom) of Mn. Since n ≥ 3, there are distinct atoms(coatoms) b, cof Mn such that b 6= a and c 6= a; so, the reflexitivity of kerh implies that(0 ∨ b, a ∨ b) = (b, 1) ∈ kerh; hence, (b ∧ c, 1 ∧ c) = (0, c) ∈ kerh which impliesthat h(c) = h(0) = h(a). Since h is a lattice homomorphism, h(1) = h(a ∨ c) =h(a)∨h(c) = h(0) which implies that h is constant; again a contradiction. Thus,h−1(h(0)) = {0} which implies that h is 0-seperating. By a similar argument, his 1-seperating. So, Mn is tight. 2

3.12 Proposition Let f : L → M be a non-constant homomorphism froma lattice L into a lattice M . If L is 0-1-simple, then f(L) is also 0-1-simple.

Proof: It is clear that f(L) is a sublattice of M . We show thatf(L) is 0-1-simple. Let M ′ be a lattice and let g : f(L) → M ′ be a non-constant homomorphism. Next, we show that g is 0-1-separating. Obviously,gf : L → M ′ is a non-constant homomorphism. Since L is 0-1-simple, gf is 0-1-separating. So, (gf)−1(gf)(0) = {0} and (gf)−1(gf)(1) = {1}; and hence, {0} =f−1g−1(g(f(0))) = f−1g−1(g(0)) and {1} = f−1g−1(g(f(1))) = f−1g−1(g(1)).Since f is surjective, {0} = f({0}) = f(f−1g−1(g(f(0)))) = f(f−1g−1(g(0))) =g−1(g(0)) and {1} = f({1}) = f(f−1g−1(g(f(1))) = f(f−1g−1(g(1))) =g−1(g(1)); and so g is 0-1-separating. 2

34

Let L and M be isomorphic lattices. If L has no non-constant stronglyextensive meet-endomorphism, the definitions of composition and strongly ex-tensive meet-endomorphism imply that M has also no non-constant stronglyextensive meet-endomorphism.

We now give some sufficient conditions for an induced algebra to betame.

3.13 Theorem Let A be a finite algebra and U = e(A) for somee ∈ E(A).

(i) If Con(A) is tight, then A|U is tame.

(ii) If Con(A) ∼= Mn, for some n ≥ 3, then A|U is tame.

(iii) If U ∈ Min(A), then A|U is tame.

Proof: Let A be an algebra and let e be an idempotent unary polyno-mial operation of A such that U = e(A).

(i) Assume that Con(A) is tight. Then, by Proposition 2.15,Con(A|U ) is tight. Hence, Theorem 3.10 implies that A|U is tame.

(ii) is a consequence of Propositions 3.11 and 3.12.

(iii) Assume that U ∈ Min(A). Then Theorem 2.10 implies thatall non-constant unary polynomial operations of A|U are bijective which implythat U is a minimal set of A|U . It is clear that U satisfies Conditions (ii) and(iii) of Definition 3.1. Since e|U is an identity mapping on U , we have that Uis an image of an idempotent operation of A|U which implies that U satisfiesCondition (i) of Definition 3.1. Thus, A|U is tame. 2

Now, we have a necessary condition of a unary tame algebra.

3.14 Proposition Let A = (A; f) be a unary algebra. If A is tame, thenf is either surjective or non-idempotent.

Proof: We assume that f is a non-surjective idempotent with U =f(A) ⊂ A. Then there is an element a ∈ A such that a 6∈ U but f(a) ∈ U .Let θ := ∆A ∪ {(a, f(a)), (f(a), a)}. It is clear that θ is an equivalence relationon A. To show that θ is compatible with f , let (x, y) ∈ θ. If x = y, then(f(x), f(y)) ∈ θ. We consider the case (x, y) 6∈ ∆A. Then (x, y) is either(a, f(a)) or (f(a), a). Since f is idempotent, (f(x), f(y)) = (f(a), f(a)) ∈ θ. Itis clear that θ|U = ∆U . Thus, A is not tame. 2

Now, we have necessary and sufficient conditions for a tame algebra tobe simple.

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3.15 Proposition Let A be an algebra.

(i) If A is simple, then A is tame.

(ii) If A is tame and |V | = 2 for some V ∈ Min(A), then A issimple.

Proof: Let A be an algebra.

(i) We show that Con(A) = {∆A, A × A} is tight. Let f be astrongly extensive meet-endomorphism. Then f(∆A) = A×A and f(A×A) =A×A which imply that f is constant. So, Con(A) has no non-constant stronglyextensive meet-endomorphism. Let L be a lattice and let f : Con(A) → L be anon-constant homomorphism. Then f(∆A) 6= f(A × A) which implies that fis injective. So, f−1f(∆A) = {∆A} and f−1f(A× A) = {A × A} which implythat f is 0-1-separating. So, Con(A) is 0-1-simple which implies that Con(A)is tight. Therefore, A is tame.

(ii) We assume that A is not simple and there is a minimal setV such that |V | = 2. Then there exists a congruence θ ∈ Con(A) such that∆A ⊂ θ ⊂ A× A. If θ satisfies Condition (ii) of Definition 3.1, then ∆V ⊂ θ|V ;so, there are distinct elements a, b ∈ V such that (a, b) ∈ θ ∩ V 2 = θ|V . Since|V | = 2, we have V = {a, b} which implies that θ|V = V × V . Hence, θ does notsatisfy Condition (iii) of Definition 3.1. If θ satisfies Condition (iii) of Definition3.1, then a similar argument implies that θ does not satisfy Condition (ii) ofDefinition 3.1. Thus, A is not tame. 2

The following proposition is obvious.

3.16 Proposition Let A and B be algebras having the same set of uni-verse. If A and B are polynomially equivalent, then

(i) Con(A) = Con(B).

(ii) Min(A) = Min(B).

(iii) A is tame if and only if B is tame.

3.17 Corollary Let B be a subalgebra of an algebra A. Then

B is tame if and only if A|B is tame.

Proof: Let B be a subalgebra of an algebra A. Then B is closed withrespect to all polynomial operations of A. So, P (A)|B = P (B) which impliesthat A|B and B are polynomailly equivalent. Hence, the corollary follows fromProposition 3.16. 2

We close the chapter by introduce the concept of type which will alsoplay an important role in Tame Congruence Theory, especially for characterizingall minimal algebras.

36

3.18 Definition [5] Let M be a minimal algebra. Then we define the typeof M as follows :

M has type 1 ⇐⇒ M is polynomially equivalent to an algebra

(M ; π) for some π ⊆ Sm (unary type)

M has type 2 ⇐⇒ M is polynomially equivalent to a vector

space (vector space)

M has type 3 ⇐⇒ M is polynomially equivalent to B =

({0, 1};∧,¬) (Boolean type)

M has type 4 ⇐⇒ M is polynomially equivalent to L =

({0, 1};∧,∨) (lattice type)

M has type 5 ⇐⇒ M is polynomially equivalent to a two-element

semilattice (semilattice type).

The following theorem about the type of a finite algebra was provedin [5].

3.19 Theorem [5] A finite algebra A is a minimal algebra if and only ifit is of type i for some i ∈ {1, 2, 3, 4, 5}.

In [5], K. Denecke characterized all minimal algebras via the conceptof types.

3.20 Theorem [5] Let A be a minimal algebra. Then

1. M has type 3 if and only if V (M) is congruence distributive andthere is a n ≥ 2 such that V (M) is n-permutable.

2. M has type 4 if and only if V (M) is congruence distributive andthere is no a n ≥ 2 such that V (M) is n-permutable.

3. M has type 5 if and only if V (M) is not congruence distributiveand there is no a n ≥ 2 such that V (M) is n-permutable, but V (M) is meet-endomorphism.

4. M has type 1 if and only if V (M) is not congruence distributiveand there is a n ≥ 2 such that V (M+) is n-permutable, but V (M) is not meet-endomorphism.

5. M has type 2 if and only if V (M) is not both congruence dis-tributive and meet-endomorphism and there is no a n ≥ 2 such that V (M+) isn-permutable.

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Theorem 3.20 characterizes all minimal algebras by the concept oftypes. We want to extend the classification of types to all finite algebras via typesof minimal algebras. As we have seen in Theorem 3.4 that minimal algebrasinduced by a tame algebra are isomorphic to each other, we can use a type of aminimal algebra to assign a type of the belonging algebra.

3.21 Definition [5] Let A be a finite tame algebra. If one of the minimalalgebra of A is of type i, i ∈ {1, 2, 3, 4, 5}. Then A is said to be of type i andwe write type A := i. We denote the set of all types of A by type {A}.

Finally, we will introduce some properties of tame algebras which aresufficient properties for algebras to be type 1 or 2.

3.22 Definition [5] An algebra A satisfies the strong term condition if forall n ∈ N, every n-ary term operation tA of A and all b, c1, . . . , cn, d1, . . . , dn ∈ Athe following implication is satisfied

tA(c1, . . . , cn) = tA(d1, . . . , dn) ⇒ tA(b, c2, . . . , cn) = tA(b, d2, . . . , dn)

An algebra satisfying the strong term condition is called strongly abelian.

3.23 Definition [5] An algebra A satisfies the term condition if for all n ∈N, every n-ary term operation tA of A and all a, b, c2, . . . , cn, d2, . . . , dn ∈ A thefollowing implication is satisfied

tA(a, c2, . . . , cn) = tA(a, d2, . . . , dn) ⇒ tA(b, c2, . . . , cn) = tA(b, d2, . . . , dn)

An algebra which satisfies the term condition is called abelian.

It is clear that if an algebra A satisfies the strong term condition thenit also satisfies the term condition, so that strongly abelian algebras are abelian.

3.24 Example Consider the binary operations f and g which aregiven by Tables 3.1

f 0 1 2 30 0 0 1 11 0 0 1 12 0 0 1 12 2 2 3 3

g 0 1 2 30 0 0 1 11 0 0 1 12 0 0 1 12 2 2 0 0.

Table 3.1

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It can be verified that f satisfies the implication of the strong term conditionand g satisfies the implication of the abelian condition. So, ({0, 1, 2, 3}; f) is astrongly abelian algebra and ({0, 1, 2, 3}; g) is an abelian algebra. ¯

The following theorem which was proved in [5] characterizes all typesof abelian and strongly abelian algebras.

3.25 Theorem [5] Let A be a tame algebra. Then

(i) A is strongly abelian if and only if type A = 1

(ii) A is abelian if and only if type A ∈ {1, 2}.

Chapter 4

Types of Order-primal Algebras

Recall that a clone C on a set A is a set of operations defined on Awhich is closed under composition and contains all projections and O(A) is theclone of all operations defined on A.

Let A be an algebra. Recall that T (A) is the clone of all term oper-ations of A. A finite algebra A is called primal if T (A) = O(A); that is, everyoperation defined on A is a term operation of A.

Let ≤ be a binary relation on A. Then ≤ is called order on a set A if≤ is reflexive, anti-symmetric and transitive. For an order ≤, we often prefer towrite x ≤ y in place of (x, y) ∈≤. Given an order ≤ on a nonempty set A, thepair (A;≤) is called an ordered set. Let (A;≤) be a finite ordered set. We denotePol(A) to be the set of all operations defined on A which preserve the order ≤.Then a finite algebra A = (A; FA) is called order-primal if T (A) = Pol(A).

An ordered set (A;≤) is called connected if for any two elements a, b ∈A there exist a natural number n and elements a = a0, a1, . . . , an = b such thateither a0 ≤ a1 ≥ a2 ≤ . . . ≥ an(≤ an) = b or a0 ≥ a1 ≤ a2 ≥ . . . ≤ an(≥ an) = b.Clearly, if (A;≤) has the least element or the greatest element then (A;≤) isconnected. An ordered set (A;≤) is called an antichain if every pair of elementsin A are non-comparable.

In this chapter, we will investigate some properties of an order-primalalgebra via Tame Congruence Theory. In Chapter 3, we stated that an algebraA is abelian if and only if type A ∈ {1, 2}. We will prove that every order-primalalgebra A is not abelian which implies that type A 6∈ {1, 2}. Moreover, we willcharacterize all order-primal algebras whose types are 3 (Boolean type) andall order-primal algebras whose type are 4 (lattice type).

As we have seen in Chapter 4 that types of algebras can be extened toconsider in varieties; so we will also find all possible types of a variety generatedby an order-primal algebra. Finally, we will give some sufficient conditions forthe variety generated by an order-primal algebra to be residually small.

39

40

We will first show that every order-primal algebra A has a minimal setwhose cardinality is 2.

4.1 Theorem Let A be an order-primal algebra corresponding to theordered set (A;≤). Then A has a minimal set whose cardinality is 2.

Proof: Let A be an order-primal algebra corresponding to the orderedset (A;≤). We show that there is a unary polynomial f of A such that f(A)satisfies the conditions (i) and (ii) of Definition 2.7.

If (A;≤) is an antichain, then every operation on A is order-preserving.Since |A| ≥ 2, there are elements a, b ∈ A such that a 6= b. We define a unaryoperation f on A by

f(t) =

{a, x = ab, otherwise.

Clearly, f is order-preserving; and so, f ∈ P (A) and f(A) is a minimal set of A.

If (A;≤) is not an antichain, then there are distinct elements a, b ∈ Asuch that a ≤ b. We define a unary operation g on A by

g(x) =

{a, x ≤ ab, otherwise.

To prove that g is order-preserving with respect to ≤, let x, y ∈ A such thatx ≤ y. If x ≤ a, then g(x) = a and g(y) ∈ {a, b} which imply that g(x) ≤ g(y).If x 6≤ a, then y 6≤ a which implies that g(x) = b = g(y). So, g ∈ P (A) andg(A) = {a, b} is a minimal set of A. 2

Lemma 4.2 was proved by B.A. Davey in [3].

4.2 Lemma [3] Let A be an order-primal algebra corresponding to theordered set (A;≤). Then

(i) A has no proper subalgebras.

(ii) If (A;≤) is either connected or an antichain, then A is simple.

(iii) If (A;≤) is neither connected nor an antichain, then A hasonly one proper nontrivial congruence.

If an order-primal algebra is tame we call it a tame order-primal alge-bra. The following proposition which is a consequence of Proposition 3.15 andLemma 4.2, characterizes all tame order-primal algebras.

4.3 Proposition Let A be an order-primal algebra corresponding to theordered set (A;≤). Then A is tame if and only if (A;≤) is either connected oran antichain.

41

We have the following corollary.

4.4 Corollary If A is a tame order-primal algebra corresponding to theordered set (A;≤), then |V | = 2 for all V ∈ Min(A).

Proof: Let V be a minimal set of an order-primal algebra A. Then A|Vis a minimal algebra of A. By Theorem 4.1, there is a minimal set U of A with|U | = 2. Since A is tame, Lemma 3.3 and Theorem 3.4 imply that A|V and A|Uare isomorphic; so, |V | = |U | = 2. 2

We have the following proposition.

4.5 Proposition Let A be an order-primal algebra corresponding to theordered set (A;≤). Then P (A) = T (A).

Proof: Let A be an order-primal algebra corresponding to the orderedset (A;≤). Then every constant operation of A is order-preserving which impliesthat {ca | a ∈ A} ⊆ Pol(A) = T (A); so, P (A) = < T (A) ∪ {ca | a ∈ A} > =<T (A) >. Since T (A) is a subclone of P (A) and P (A) is the smallest clonecontaining T (A), we have P (A) = T (A). 2

The following proposition let us to see all of minimal sets of an order-primal algebra corresponding to a connected ordered set.

4.6 Proposition Let A be an order-primal algebra corresponding to theordered set (A;≤). If (A;≤) is connected, then subordered set (V ;≤) is a two-element lattice for all V ∈ Min(A).

Proof: Let A be an order-primal algebra corresponding to the con-nected ordered set (A;≤). Then A is tame. Let V be a minimal set of A. Then|V | = 2 and there is an idempotent unary polynomial e of A such that e(A) = V .We may assume that V = {a, b}.

Suppose that a, b are non-comparable. Then there are elements a =x0, . . . , xn = b ∈ A such that x0 ≤ x1 ≥ x2 ≤ . . . ≥ xn or x0 ≥ x1 ≤ x2 ≥ . . . ≤xn. So, xi and xi+1 are comparable for all i = 0, . . . , n. To show that e(x) = e(y)for all pairs x, y ∈ A which are comparable, suppose that there are distinctelements x, y ∈ A such that x, y are comparable and e(x) 6= e(y). If e(x) = a,then e(y) = b which implies that e(x) and e(y) are non-comparable. But e isorder-preserving. So, e(x) and e(y) are comparable which is a contradiction.One can get a similar contradiction if e(x) = b and e(y) = a. So, e(xi) = e(xi+1)for all i = 0, . . . , n; and hence, e(a) = e(x0) = e(x1) = . . . = e(xn) = e(b); hence,a = b. So, |V | = 1, a contradiction. Thus, (V ;≤) is a two-lattice. 2

The following theorem which was proved in [5] shows that the propertyof an abelian algebra can be characterized by properties of the congruence ofthe direct product of the algebra.

42

4.7 Theorem [5] An algebra A is abelian if and only if there is a con-gruence relation θ on A × A such that the diagonal ∆A = {(a, a) | a ∈ A} is acongruence class of θ.

The following theorem is useful for characterizing all types of order-primal algebra.

4.8 Theorem Every order-primal algebra is not abelian.

Proof: Let A be an order-primal algebra corresponding to the or-dered set (A;≤). Suppose that A is abelian. Then there is a congruenceθ ∈ Con(A× A) such that ∆A = {(a, a) | a ∈ A} is a congruence class of θ.

If (A;≤) is an antichain, every operation on A is order-preserving.Since |A| ≥ 2, there are distinct elements a, b ∈ A. We consider the binaryoperation f : A2 → A which is defined by

f(x, y) =

{a, x = a or y = ab, otherwise.

Then f ∈ P (A). Let F be an operation of A×A induced by f . Since (a, a) θ(b, b)and (a, b) θ(a, b), we have

F ((a, a), (a, b)) θ F ((b, b), (a, b))

which implies that (f(a, a), f(a, b)) θ (f(b, a), f(b, b)).

So, (a, a) θ(a, b) implies that (a, b) ∈ ∆A, a contradiction.

If (A;≤) is not an antichain, there are distinct elements a, b ∈ A suchthat a ≤ b. We define the following binary operation g: A2 → A by

g(x, y) =

{a, x ≤ a and y ≤ ab, otherwise.

To show that g is order-preserving, let x ≤ u and y ≤ v. If x ≤ a and y ≤ athen g(x, y) = a and g(u, v) ∈ {a, b} imply that g(x, y) ≤ g(u, v). If x 6≤ aor y 6≤ a then u 6≤ a or v 6≤ a which imply that g(x, y) = b = g(u, v). Thus,g ∈ P (A). Let G be an operation of A × A induced by g. Then, (a, a) θ(b, b)and (a, b) θ(a, b) imply that

G((a, b), (a, a)) θ G((a, b), (b, b)).

So, (g(a, a), g(b, a)) θ (g(a, b), f(b, b)); hence, (b, a) θ(b, b). Therefore, (b, a) ∈ ∆A

which is a contradiction. 2

We have the following corollary.

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4.9 Corollary Let A be an order-primal algebra corresponding to theordered set (A;≤). Then type A 6∈ {1, 2}.


4.10 Theorem [5] Every primal algebra has type 3 (Boolean type).

We have the following propositions.

4.11 Proposition Let A be a primal algebra and U ⊆ A. Then theinduced algebra A|U is also primal.

Proof: Let A|U be an induced algebra of a primal algebra A. We showthat P (A)|U = O(U). Clearly, P (A)|U ⊆ O(U). Assume that f be n-ary oper-ation on U . Since A 6= ∅, there is an element c ∈ A. We define the followingn-ary operation f : An → A by

f(a1, . . . , an) =

{f(a1, . . . , an), a1, . . . , an ∈ U

c, otherwise.

It is clear that f ∈ P (A), f|U = f and f(Un) ⊆ U . So, f = f|U ∈ P (A)|U . 2

The following proposition shows that every primal algebra has uniquetype.

4.12 Proposition Let A be a primal algebra. Then type {A} = {3}.

Proof: Let A be a primal algebra. Then Theorem 4.10 implies that Ahas type 3.

Suppose that A is of type 4. Then, there is a two-element minimalalgebra M of A such that M is of type 4. We assume that M = {0, 1}. SinceA is primal, M is also primal; so, T (M) = O(M). Hence, every projectionoperation on {0, 1} is a term operation of M and there are term operations∧,∨,¬ and · which satisfy

0 ∧ 0 = 0 ∧ 1 = 1 ∧ 0 = 0 and 1 ∧ 1 = 1

0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1 and 0 ∨ 0 = 0

0 · 0 = 0 · 1 = 1 · 0 = 0 and 1 · 1 = 1

¬(0) = 1 and ¬(1) = 0. (4.1)

Theorem 3.20 implies that the variety V (M) is a congruence distributive andthere is no n ≥ 2 such that V (M) is n-permutable. Let t(x, y, z) be a ternaryoperation on {0, 1} defined by Table 4.1.

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x 0 0 0 0 1 1 1 1y 0 0 1 1 0 0 1 1z 0 1 0 1 0 1 0 1

t(x, y, z) 0 1 0 0 1 1 0 1.

Table 4.1

It is clear that t is a term operation of M and it can be expressed as t(x, y, z) =(z ∧ (xy ∨ (¬x)(¬y))) ∨ (x ∧ ¬y) for all x, y, z ∈ {0, 1}. So, M satisfies thefollowing identities

e3,A1 (x, y, z) = x

e3,A1 (x, x, y) = t(x, y, y)

t(x, x, y) = e3,A3 (x, y, y)

e3,A3 (x, y, z) = z (4.2)

which implies that V (M) satisfies (4.2). Theorem 1.6.7 implies that V (M) is2-permutable which is a contradiction. Thus, 4 6∈ type {A}.

Suppose that A is of type 5. Then, there is a two-element minimalalgebra M of A such that M is of type 5. We suppose that M = {0, 1}. SinceM is primal, M satisfies (4.1). By Theorem 3.20, the variety V (M) is notcongruence distributive and there is no n ≥ 2 such that V (M) is n-permutable.Let p(x, y, z) be a ternary operation on {0, 1} defined by Table 4.2.

x 0 0 0 0 1 1 1 1y 0 0 1 1 0 0 1 1z 0 1 0 1 0 1 0 1

p(x, y, z) 0 0 0 1 0 1 1 1.

Table 4.2

It is clear that p is a term operation of M and it can be expressed as p(x, y, z) =(x∧y)∨ (x∧z)∨ (y∧z) for all x, y, z ∈ {0, 1}. Next we will show that p satisfiesthe identities in Theorem 1.6.4. By using the definition of p, we have

p(x, x, y) = (x ∧ x) ∨ (x ∧ y) ∨ (x ∧ y) = x ∨ (x ∧ y) = x

p(x, y, x) = (x ∧ y) ∨ (x ∧ x) ∨ (y ∧ x) = x ∨ (x ∧ y) = x

p(y, x, x) = (y ∧ x) ∨ (y ∧ x) ∨ (x ∧ x) = (y ∧ x) ∨ x = x

So, M satisfies the identities of Theorem 1.6.4 which implies that V (M) satisfiesthose identities; hence, V (M) is congruence distributive, a contradiction. Thus,5 6∈ type {A}. 2

We will characterize all tame order-primal algebras whose type is 3.

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4.13 Theorem Let A be a tame order-primal algebra correspondingto the ordered set (A;≤). Then (A;≤) is an antichain if and only if type{A} = {3} (Boolean type).

Proof: Let A be a tame order-primal algebra corresponding to the an-tichain (A;≤). Then every operation on A is an order-preserving mapping withrespect to ≤ which implies that T (A) = Pol(A) = O(A); that is, A is primal.So, Proposition 4.12 implies that type {A} = {3}.

Conversely, we let A be an order-primal algebra corresponding to theordered set (A;≤) which is of type 3. Suppose that (A;≤) is connected. Then,there is a two-element minimal algebra M of A such that V (M) is congru-ence distributive and there is a integer n ≥ 2 such that V (M) is n-permutableand (M ;≤) is a two-element lattice. So, M is n-permutable. We assume thatM = {0, 1}. Definition 3.18 implies that M is polynomially equivalent to aBoolean algebra B = ({0, 1};∧,¬). So, B is n-permutable. By Theorem 1.6.8,B is 2-permutable or 3-permutable which implies that M is 2-permutable or3-permutable. If M is 2-permutable, then M is 3-permutable and there areternary terms p0, p1, p2 such that

p0(x, y, z) = x

p0(x, x, y) = p1(x, y, y)

p1(x, x, y) = p2(x, y, y)

p2(x, y, z) = z

So, p1(x, y, y) = x and p1(x, x, y) = y. By 0 ≤ 0, 0 ≤ 1 and 1 ≤ 1, we have1 = p1(0, 0, 1) ≤ p1(0, 1, 1) = 0 which is a contradiction.

If M is 3-permutable but is not 2-permutable, then there are ternaryterms p0, p1, p2 and p3 such that

p0(x, y, z) = x

p0(x, x, y) = p1(x, y, y)

p1(x, x, y) = p2(x, y, y)

p2(x, x, y) = p3(x, y, y)

p3(x, y, z) = z

We show that p2(x, y, y) = x for all x, y ∈ {0, 1}. Suppose that there are elementsx, y ∈ {0, 1} such that p2(x, y, y) 6= x. If x = y, then p2(x, y, y) = p2(x, x, x) = xwhich is a contradiction. So, x 6= y; hence, p2(x, y, y) = y which implies thatp1(x, x, y) = p2(x, y, y) = p3(x, y, y) = y. Therefore, the operation p0, p1 and p3

satisfy the following identities

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p0(x, y, z) = x

p0(x, x, y) = p1(x, y, y)

p1(x, x, y) = p3(x, y, y)

p3(x, y, z) = z

which implies that A is 2-permutable, a contradiction. So, p2(x, y, y) = x andp2(x, x, y) = y for all x, y ∈ {0, 1}. Since 0 ≤ 0, 0 ≤ 1 and 1 ≤ 1, we have1 = p2(0, 0, 1) ≤ p2(0, 1, 1) = 0 which is again a contradiction. Thus, (A;≤) isan antichain. 2

The following propositions are useful for the proof of Theorem 4.15.

4.14 Proposition Let A be an order-primal algebra corresponding to theordered set (A;≤) and U ⊆ A. Then

(i) P (A|U ) ⊆ Pol(U), and

(ii) if (A;≤) is an antichain and |U | = 2, then A|U is an order-primal algebra corresponding to the subordered set (U ;≤).

Proof: Let A be an order-primal algebra corresponding to (A;≤).

(i) Let A|U be an induced algebra of A and h be a n-ary polynomialoperation of A|U . Then there is a n-ary order-preserving mapping hA on A suchthat hA|U = h and h(Un) ⊆ U which imply that h is an order-preserving mappingon U . So, h ∈ Pol(U).

(ii) Let (A;≤) be an antichain and let U be a subset of A whosecardinality is 2. We show that P (A|U ) = Pol(U). Then (i) implies that P (A|U ) ⊆Pol(U). To show that Pol(U) ⊆ P (A|U ), let h be a n-ary order-preservingmapping on U and (b1, ..., bn) ∈ An. We define the following n-ary operationg : An → A by

g(a1, ..., an) =

{h(a1, ..., an), (a1, ..., an) ∈ Un

(b1, ..., bn), otherwise.

Then h ∈ P (A); so, h = g|U ∈ P (A|U ). Hence, P (A|U ) = Pol(U) which impliesthat A|U is an order-primal algebra. 2

Now, we are ready to characterize all tame order-primal algebra Awhose type is 4.

4.15 Theorem Let A be a tame order-primal algebra correspondingto the ordered set (A;≤). Then (A;≤) is connected if and only if type {A} = {4}(lattice type).

47

Proof: Let A be a tame order-primal algebra corresponding to the con-nected ordered set(A;≤). Then Corollary 4.9 and Theorem 4.13 imply that type{A} ⊆ {4, 5}. We show that 5 6∈ type {A}. Let M be a minimal algebra A.Then M is a two-element algebra such that (M :≤) is a two-element lattice andtype {M} ⊆ {4, 5}. We may assume that M = {0, 1}.

Suppose that M is of type 5. Then, there is a two-element minimalalgebra M of A such that M is of type 5 and (M ;≤) is a two-element lattice;hence, Theorem 3.20 implies that the variety V (M) is not a congruence distribu-tive and there is no n ≥ 2 such that V (M) is n-permutable. Let M = {0, 1}and let define t: A3 → A for each (x, y, z) ∈ A3 by

t(x, y, z) =

{1, at least two of {x, y, z} belongs to ↑ 10, otherwise.

To show that t is an order-preserving with respect to ≤, let a ≤ x, b ≤ y andc ≤ z. If at least two of {a, b, c} belongs to ↑ 1, then at least two of {x, y, z}belongs to ↑ 1; so, t(a, b, c) = 1 = t(x, y, z). Otherwise, t(a, b, c) = 0 andt(x, y, z) ∈ {0, 1}; hence, t(a, b, c) ≤ t(x, y, z). So, t ∈ P (A); hence, t|M ∈ P (M)and t|M is defined by Table 4.3.

x 0 0 0 0 1 1 1 1y 0 0 1 1 0 0 1 1z 0 1 0 1 0 1 0 1

t(x, y, z) 0 0 0 1 0 1 1 1.

Table 4.3

Since t(0, 0, 1) = t(0, 1, 0) = t(1, 0, 0) = 0 and t(1, 1, 0) = t(1, 0, 1) =t(0, 1, 1) = 1, we have t(x, x, y) = t(x, y, x) = t(y, x, x) = x for all x, y ∈ {0, 1};hence, M satisfies the identities of Theorem 1.6.4 which implies that V (M) alsosatisfies those identities. Therefore, V (M) is congruence-distributive, a contra-diction. Thus, type M = {4} which implies that type {A} = {4}.

Conversely, Theorem 4.13 implies that (A;≤) is not an antichain. ButA is tame; so, (A;≤) is connected. 2

4.16 Corollary Let A be an order-primal algebra corresponding tothe connected ordered set (A;≤) and U ∈ Min(A). Then the minimal algebraA|U is also an order-primal algebra corresponding to the subordered set (U ;≤).

Proof: Let A be an order-primal algebra corresponding to the con-nected ordered set (A;≤) and U ∈ Min(A). Then (U ;≤) is a lattice andA|U is polynomially equivalent to a lattice algebra L = ({0, 1};∧,∨). So,

48

T (A|U ) = P (A|U ) =< {∧,∨, c0, c1} >= Pol(U); hence, A|U is an order-primalalgebra corresponding to the (U ;≤). 2

In Tame Congruence Theory, the definition of type can be extened tovarieties. The type of a variety V of algebras is the set of all the types of finitealgebras in the variety; that is,

type {V } =⋃{type A |A ∈ V (A) and A is finite}.

D. Hobby and R. McKenzie[8] proved that there is a close connection betweenthe types of a finite algebra A and Mal’cev type conditions for the varietiesgenerated by A.

The following Theorem was proved by D. Hobby and R. McKenziewhich characterize the types which cannot occur for a variety by means of theMal’cev condition properties of the variety.

4.17 Theorem [5] Let V (A) be a variety generated by an algebra A.

(i) If V (A) is congruence distributive, then type {V (A)} ∩{1, 2, 5} = ∅.

(ii) If V (A) is congruence permutable, then type {V (A)} ⊆ {2, 3}.(iii) V (A) is n-permutable for some n ≥ 2 if and only if type

{V (A)}⊆ {2, 3}.(iv) type {V (A)} = {3} if and only if in V (A) there are terms

f0(x, y, z, u), . . . , fn(x, y, z, u) for n ≥ 2 such that the following identities holdin V (A)

f0(x, y, y, z) = x

fi(x, x, y, x) = fi+1(x, y, y, x) for all i < n

fi(x, x, y, y) = fi+1(x, y, y, y) for all i < n

fn(x, x, y, z) = z.

(v) type {V (A)} ∩ {1, 2} = ∅ if and only if V (A) is meet-semidistributive if and only if the class of all lattices isomorphic to a sublatticeof Con(B) for some B ∈ V (A) does not contain the lattice Mn.

The following lemma which was proved in [5] is useful for finding pos-sible types of a variety.

4.18 Lemma [5] Let A be a finite algebra. If V (A) is congruence mod-ular, then type {V (A)} = type {S(A)}.

The following was proved in [7].

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4.19 Theorem [7] Let V be a locally finite variety. Then Con(A) is join-semidistributive for every finite A ∈ V if and only if type {V (A)} ⊆ {3, 4}.

Now, we will introduce the concept of product congruence which alsoplay an important role for finding all possible types of a variety generated by anorder-primal algebra.

4.20 Theorem Let {Ai | i ∈ I} be a family of algebras and let θi ∈Con(Ai), i ∈ I. Then the product congruence Πi∈Iθi of Πi∈IAi is defined by

(a, b) ∈ Πi∈Iθi ⇐⇒ (πi(a), πi(b)) ∈ θi for all i ∈ I

is a congruence relation on Πi∈IAi.

Proof: Let θi ∈ Con(Ai) for all i ∈ I. It is clear that Πi∈Iθi is reflexiveand symmetric. To show that Πi∈Iθi is transitive, let (a, b) and (b, c) ∈ Πi∈Iθi.Then (πi(a), πi(b)) and (πi(b), πi(c)) ∈ θi for all i ∈ I which imply that(πi(a), πi(c)) ∈ θi for all i ∈ I. So, (a, c) ∈ Πi∈Iθi.

Finally, we show that Πi∈Iθi compatible with all unary polynomialoperations of Πi∈IAi. Let fΠi∈IAi be a unary polynomial operation of Πi∈IAi

and let (a, b) ∈ Πi∈Iθi. Then (πi(a), πi(b)) ∈ θi for all i ∈ I and

fΠi∈IAi(a) = (fAi(πi(a)))i∈I and fΠi∈IAi(b) = (fAi(πi(b)))i∈I

which imply that (fAi(πi(a)), fAi(πi(b))) ∈ θi for all i ∈ I. But

fAi(πi(a)) = πi(fΠi∈IAi(a)) and fAi(πi(b))) = πi(f

Πi∈IAi(a))

for all i ∈ I. So, (πi(fΠi∈IAi(a)), πi(f

Πi∈IAi(b))) ∈ θi for all i ∈ I which implythat (fΠi∈IAi(a), fΠi∈IAi(b)) ∈ Πi∈Iθi. 2

We denote the set of all product congruences of Πi∈IAi byΠi∈ICon(Ai).

4.21 Proposition Let {Ai | i ∈ I} be a family of algebras and let θi, βi ∈Con(Ai) for all i ∈ I. Then

(i) Πi∈I(θi) ∨ Πi∈I(βi) = Πi∈I(θi ∨ βi)

(ii) Πi∈I(θi) ∧ Πi∈I(βi) = Πi∈I(θi ∧ βi).

Proof: Let i ∈ I and θi, βi ∈ Con(Ai).

(i) Let (a, b) ∈ Πi∈I(θi) ∨ Πi∈I(βi). Then there are elementsc1, . . . , ck ∈ Πi∈IAi such that (a, c1) ∈ Πi∈I(βi), (c1, c2) ∈ Πi∈I(θi), . . . , (ck, b) ∈Πi∈I(θi). So,

50

(πi(a), πi(c1)) ∈ βi, (πi(c1), πi(c2)) ∈ θi, . . . , (πi(ck), πi(b)) ∈ θi.

for all i ∈ I; hence, (πi(a), πi(b))) ∈ θi ∨ βi for all i ∈ I which imply that(a, b) ∈ Πi∈I(θi∨βi). Conversely, let (a, b) ∈ Πi∈I(θi∨βi). Then (πi(a), πi(b))) ∈θi ∨ βi for all i ∈ I. So, for each i ∈ I, there are elements ci

1, . . . , cik ∈ Ai such

that (πi(a), ci1) ∈ βi, (c

i1, c

i2) ∈ θi, . . . , (c

ik, πi(b)) ∈ θi. For each 1 ≤ j ≤ k, let

cj ∈ Πi∈IAi be such that πi(cj) = cij for all i ∈ I. It is clear that (a, c1) ∈

β, (c1, c2) ∈ θ, . . . , (ck, b) ∈ θ. So, (a, b) ∈ θ ∨ β.

(ii) Let (a, b) ∈ θ ∧ β. Then (a, b) ∈ θ, (a, b) ∈ β. So,(πi(a), πi(b))) ∈ θi and (πi(a), πi(b))) ∈ βi for all i ∈ I; hence, (πi(a), πi(b))) ∈θi ∧ βi for all i ∈ I. Therefore, (a, b) ∈ Πi∈I(θi ∧ βi). By a similar argument,Πi∈I(θi ∧ βi) ⊆ θ ∧ β. 2

Recall that an algebra A is join-semidistributive if whenever congru-ences θ, β and φ ∈ Con(A) satisfy θ∨β = θ∨φ, they also satisfy θ∨β = θ∨(β∧φ).A variety V is join-semidistributive if all algebras in V are join-semidistributive.

The following proposition give a sufficient condition for a direct productof algebras to be join-semidistributive.

4.22 Proposition Let {Ai | i ∈ I} be a family of join-semidistributivealgebras. If Con(Πi∈IAi) = Πi∈ICon(Ai), then Πi∈IAi is join-semidistributive.

Proof: Let θ, β, φ ∈ Con(Πi∈IAi) such that θ ∨ β = θ ∨ φ. Thenθ = Πi∈Iθi, β = Πi∈Iβi and φ = Πi∈Iφi for some θi, βi, φi ∈ Con(A). Proposition4.21 implies that θ ∨ β = Πi∈I(θi ∨ βi) and θ ∨ φ = Πi∈I(θi ∨ φi). To showthat θi ∨ βi = θi ∨ φi for all i ∈ I, let i ∈ I and (ai, bi) ∈ θi ∨ βi. SinceAj 6= ∅ for all j ∈ I, j 6= i, there is an element aj ∈ Aj for all j ∈ I, j 6= i.Let a and b ∈ Πi∈IAi be such that πj(a) = aj for all j ∈ I , πj(b) = aj forall j 6= i and πi(b) = bi. Then (πj(a), πj(b)) ∈ θj ∨ φj for all j ∈ I, j 6= iand (πi(a), πi(b)) ∈ θi ∨ φi; so, (a, b) ∈ θ ∨ β. Since θ ∨ β = θ ∨ φ, we have(a, b) ∈ θ ∨ φ which implies that (πj(a), πj(b)) ∈ θj ∨ φj for all j ∈ I. But(πi(a), πi(b)) = (ai, bi). So, (ai, bi) ∈ θi∨φi. Thus, θi∨βi ⊆ θi∨φi. By a similarargument, we have θi∨φi ⊆ θi∨βi. Since Ai is join-semidistributive for all i ∈ I,we have θi ∨ βi = θi ∨ (βi ∧ φi) for all i ∈ I. So,

θ ∨ β = Πi∈I(θi ∨ βi)

= Πi∈I(θi ∨ (βi ∧ φi))

= Πi∈Iθi ∨ (Πi∈Iβi ∧ Πi∈Iφi)

= θ ∨ (β ∧ φ).

Thus, Πi∈IAi is join-semidistributive. 2

Let B be a subalgebra of an algebra A and let θ ∈ Con(B). It is clearthat θ := θ ∪∆A−B is a congruence relation on A such that θ|B = θ.

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4.23 Proposition All subalgebras of a join-semidistributive algebra arealso join-semidistributive.

Proof: Let θ, β, φ ∈ Con(B) such that θ ∨ β = θ ∨ φ. Then thereare congruence relations θ = θ ∪ ∆A−B, β = β ∪ ∆A−B and φ = φ ∪ ∆A−B

on A such that θ|B = θ, β|B = β and φ|B = φ. To show that θ|B ∨ β|B =(θ ∨ β)|B , let (a, b) ∈ θ|B ∨ β|B . Then there are elements c1, . . . , cn ∈ B suchthat a β|B c1 θ|B c2 β|B . . . cn θ|B b which implies that (a, b) ∈ (θ ∨ β) and (a, b) ∈B × B; so, (a, b) ∈ (θ ∨ β)|B . Assume that (a, b) ∈ (θ ∨ β)|B . Then (a, b) ∈(θ ∨ β) and (a, b) ∈ B ×B; so, (a, b) ∈ (θ ∪∆A−B) ∨ (β ∪∆A−B) which impliesthat there are elements c1, . . . , cn ∈ A such that a β c1 θ c2 β . . . cn θ b. Sincea, b ∈ B, we have c1, . . . , cn ∈ B which implies that a β c1 θ c2 β . . . cn θ b. So,(a, b) ∈ θ ∨ β = θ|B ∨ β|B . Next we show that θ ∨ β = (θ ∨ β) ∪ ∆A−B. Let(a, b) ∈ (θ ∪∆A−B) ∨ (β ∪∆A−B). Then there are elements c1, . . . , cn ∈ A suchthat (a, c1) ∈ β ∪∆A−B, (c1, c2) ∈ θ ∪∆A−B, . . . , (cn, b) ∈ θ ∪∆A−B.

If a, b ∈ B then a β c1 θ c2 β . . . cn θ b which implies that (a, b) ∈ θ∨β ⊆(θ ∨ β) ∪∆A−B = θ.

If a 6∈ B or b 6∈ B then a ∈ A − B or b ∈ A − B. Assume thata ∈ A − B. Then (a, c1) ∈ ∆A−B, (c1, c2) ∈ ∆A−B, . . . , (cn, b) ∈ ∆A−B whichimply that (a, b) ∈ ∆A−B ⊆ (θ ∨ β) ∪∆A−B.

Conversely, let (a, b) ∈ (θ ∨ β) ∪ ∆A−B. Then (a, b) ∈ θ ∨ β or(a, b) ∈ ∆A−B. If (a, b) ∈ θ ∨ β, then there are elements c1, . . . , cn ∈ Bsuch that a β c1 θ c2 β . . . cn θ b which implies that (a, c1) ∈ β ∪ ∆A−B, (c1, c2) ∈θ ∪∆A−B, . . . , (cn, b) ∈ θ ∪∆A−B. So, (a, b) ∈ (θ ∪∆A−B) ∨ (β ∪∆A−B).

If (a, b) ∈ ∆A−B, then a = b; so, (a, b) ∈ (θ ∪∆A−B) ∨ (β ∪∆A−B).

We show that θ|B ∧ β|B = (θ∧ β)|B . Let (a, b) ∈ θ|B ∧ β|B . Then (a, b) ∈θ|B and (a, b) ∈ β|B imply that (a, b) ∈ B × B; so, (a, b) ∈ θ ∧ β. Therefore,(a, b) ∈ (θ∧ β)|B . By a similar argument, we have (θ∧ β)|B ⊆ θ|B ∧ β|B . To showthat θ ∧ β = (θ ∧ β) ∪∆A−B, let (a, b) ∈ θ ∧ β. Then (a, b) ∈ θ and (a, b) ∈ βwhich imply that (a, b) ∈ θ ∧ β or (a, b) ∈ ∆A−B; so, (a, b) ∈ (θ ∧ β) ∪ ∆A−B.By a similar argument, (θ ∨ β) ∪∆A−B ⊆ θ ∧ β. Since θ ∨ β = θ ∨ φ, we have(θ ∨ β) ∪ ∆A−B = (θ ∨ φ) ∪ ∆A−B which implies that θ ∨ β = θ ∨ φ; and so,θ ∨ β = θ ∨ (β ∧ φ). Hence,

θ ∨ β = θ|U ∨ β|U= (θ ∨ β)|U= (θ ∨ (β ∧ φ))|U= θ|U ∨ (β ∧ φ)|U= θ|U ∨ (β|U ∧ φ|U )

= θ ∨ (β ∧ φ).

Thus, B is join-semidistributive. 2

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Now, we will give a sufficient condition for a variety generated by ajoin-semidistributive algebra to be join-semidistributive.

4.24 Proposition Let V (A) be a variety generated by a join-semidistributive algebra A. If Con(AI) = Con(A)I for each index set I, thenV (A) is a join-semidistributive variety.

Proof: Let B ∈ V (A). Then there are a subalgebra C of AI and ahomomorphism ϕ: C → B such that ϕ(C) = B. Proposition 4.22 implies that AI

is join-semidistributive which implies that C is join-semidistributive. We showthat ϕ(C) is join-semidistributive. By Proposition 3.6, a mapping µ: [kerϕ, C ×C] → Con(B), which is defined by µ(θ) = ϕ(θ), is a lattice isomorphism. Letθ, β and φ ∈ Con(B) such that θ∨β = θ∨φ. Then there are congruence relationsθ, β and φ ∈ [kerϕ, C × C] such that ϕ(θ) = θ, ϕ(β) = β and ϕ(φ) = φ. So,

θ ∨ β = ϕ(θ) ∨ ϕ(β) = µ(θ) ∨ µ(β) = µ(θ ∨ β) = ϕ(θ ∨ β)

θ ∨ φ = ϕ(θ) ∨ ϕ(φ) = µ(θ) ∨ µ(φ) = µ(θ ∨ φ) = ϕ(θ ∨ φ).

Since θ ∨ β = θ ∨ φ and µ is injective, θ ∨ β = θ ∨ φ which implies thatθ ∨ β = θ ∨ (β ∧ φ). So,

θ ∨ β = ϕ(θ ∨ β) = ϕ(θ ∨ (β ∧ φ)) = µ(θ) ∨ µ(β ∧ φ) = µ(θ) ∨ (µ(β) ∧ µ(φ)) =

ϕ(θ) ∨ (ϕ(β) ∧ ϕ(φ)) = θ ∨ (β ∧ φ).

Thus, V (A) is join-semidistributive. 2

R. McKenzie [9] showed that in any lattice, there are term operationsm(x, y, z), p(x, y), q(x, y) such that satisfy the following identities

m(x, y, y) = y

p(x, y, ) = p(y, x)

q(x, y, ) = q(y, x) (*)

and he proved that the congruences on the cartesian product of lattices canbe taken apart into congruences on the factor lattices by using (*). More-over, he suggested that this result hold for cartesian product of algebras whichm(x, y, z), p(x, y), q(x, y) built from variables and the fundamental operationsfor which certain identities hold in both the factor algebras. From his idea, wehave the following proposition.

4.25 Proposition Let {Ai | i ∈ I} be a family of algebras. If Ai has termoperations which satisfy (*) for all i ∈ I, then Con(Πi∈IAi) = Πi∈ICon(Ai).

The following theorem which was proved in [3].

53

4.26 Theorem [3] Let A be an order-primal algebra corresponding to theordered set (A;≤).

(i) If (A;≤) is an antichain, then A is primal; hence, V (A) iscongruence distributive.

(ii) If (A;≤) is connected, then V (A) is congruence distributive ifand only if A is the only subdirectly irreducible algebra in V (A).

The following propositions give some tools to find all possible types ofa variety generated by an order-primal algebra.

4.27 Proposition Let A be an order-primal algebra corresponding to anantichain (A;≤). Then type {V (A)} = {3}.

Proof: Let (A;≤) be an antichain. Then A is primal and type {A} ={3}. So, Theorem 4.26 implies that V (A) is congruence distributive; hence,V (A) is congruence modular. By Lemma 4.18, type {V (A)} = type {S(A)}.Since A has no non-proper subalgebra, type {V (A)} = type {A} = {3}. 2

4.28 Proposition Let A be an order-primal algebra corresponding to aconnected ordered set (A;≤).

(i) If V (A) is congruence modular, then type {V (A)} = {4}.(ii) If V (A) is not congruence modular, then type {V (A)} ⊆

{3, 4, 5}.(iii) If V (A) is join-semidistributive, then type {V (A)} ⊆ {3, 4}.(iv) If A has term operations m(x, y, z), p(x, y), q(x, y) which sat-

isfy the following identities

m(x, y, y) = y

p(x, y, ) = p(y, x)

q(x, y, ) = q(y, x),

then type {V (A)} ⊆ {3, 4}.

Proof: Let (A;≤) be connected. Then type {A} = {4}.(i) is a consequence of Lemma 4.18.

(ii) Since A is simple, the class of all lattice isomorphic of sublat-tices of Con(A) does not contain the lattice M3. So, Theorem 4.17 implies thattype {V (A)} ∩ {1, 2} = ∅; that is, type {V (A)} ⊆ {3, 4, 5}.

(iii) Since A is finite, V (A) is locally finite. So, Theorem 4.19implies that type {V (A)} ⊆ {3, 4}.

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(iv) Since A satisfies (*), Con(AI) = Con(A)I for each index setI. So, Proposition 4.24 implies that V (A) is join-semidistributive. Theorem4.19 implies that type {V (A)} ⊆ {3, 4}. 2

We close the chapter by introduce the concept of residually small va-riety which is a application of Tame Congruence Theory.

A variety V is called residually small if there is a cardinal numberλ such that every subdirectly irreducible algebra in V has at most λ element.Varieties which are not residually smalll are called residually large.

The following proposition and Theorem were proved in [5].

4.29 Proposition [5] Every variety generated by a primal algebra is residu-ally small.

4.30 Theorem [5] Every locally finite variety which omits the types 1 and5, and is residually small, is congruence-modular.

We will give some sufficient condition for a variety generated by anorder-primal algebra to be residually small.

4.31 Proposition Let A be an order-primal algebra corresponding to anantichain (A;≤). Then V (A) is residually small.

Proof: Let (A;≤) be an antichain. Then A is primal. So, Proposition4.29 implies that V (A) is residually small. 2

4.32 Proposition Let A be an order-primal algebra corresponding to aconnected ordered set (A;≤).

(i) If V (A) is congruence distributive, then V (A) is residuallysmall.

(ii) If V (A) is not congruence distributive, then V (A) is residuallylarge or 5 ∈ type {A}.

Proof: Let (A;≤) is connected.

(i) Since V (A) is congruence distributive, Theorem 4.26 impliesthat A is the only subdirectly irreducible algebra in V (A). Sine A is finite, thecardinal number λ such that every subdirectly irreducible algebra in V (A) hasat most λ to be the cardinality of A. So, V (A) is residually small.

(ii) is a consequence of Theorem 4.30. 2

55

References

[1] S. Burris and H.P. Sankappanavar(1981) A course in Universal Algebra,Graduate Texts in Mathematices 78, Springer, New York.

[2] B.A. Davey and H.A. Priestley(1990) Introduction to Lattice and Or-dered Set, Cambridge University Press, Cambridge.

[3] B.A Davey, R.W. Quackenbush, and D. Schweigert(1990) MonotoneClones and the Varieties They Determine, Order 7, 145-167.

[4] J. Demetrovice and L. Ronyai(1989) Algebraic Properties of Crowns andFences, Order 6, 91-99.

[5] K. Denecke(1998) Tame Congruence Theory, East-West Jurnal ofMathematics 1, 1-42.

[6] K. Denecke and S.L. Wismath(2001) Universal Algebra and Applicationsin Theoretical Computer Science, CRC Press, New York.

[7] R. Freese, K. Kearnes, and J.B. Nation(1995) Congruence Lattices of Con-gruence Semidistributive Algebras, Lattice Theory and its Applications, 63-78.

[8] D. Hobby and R.McKenzie(1988) The Structure of Finite Algebras(TameCongruence Theory), AMS Contemporary Mathematics Series, Providence,Rhode Island.

[9] R. McKenzie, F. McNulty, and W. Taylor(1987) Algebras, Lattices, Va-rieties. Volume 1, The Maple-Vail Book Manufacturing Group, Pennsylvania.

56

.

Appendix A

.

Lattice Theory

.

Definition 1 Let P be a set. An order(or partial order) on P is a binaryrelation ≤ on P such that for all x, y, z ∈ P ,

(i) x ≤ x,

(ii) x ≤ y and y ≤ x imply x = y,

(iii) x ≤ y and y ≤ z imply x ≤ y.

These conditions are referred to reflexive, anti-symmetry and transi-tive; respectively. A set P equipped with and order relation ≤ is said to be anordered set(or partially ordered set) and we write (P ;≤).

Definition 2 Let (P ;≤) be an ordered set. Then (P ;≤) is called a chainif and only if for all x, y ∈ P , either x ≤ y or y ≤ x.

Definition 3 Let (P ;≤) and (Q;≤) be ordered sets. A map ϕ : P → Qis said to be

(i) an order-preserving if x ≤ y in P imply ϕ(x) ≤ ϕ(y) in Q,

(ii) an order-embedding if x ≤ y in P if and only if ϕ(x) ≤ ϕ(y)in Q,

(iii) an order-isomorphism if it is an order-embedding mappingfrom P onto Q.

If there exists an order-isomorphism from P onto Q, we say that Pand Q are order-isomorphic.

Lemma 4 (i) An order-embedding is a one-to-one map.

(ii) An order-isomorphism is bijective.

Let (P ;≤) be an ordered set. For x, y ∈ P , we write x ∨ y in place ofthe least upper bound of {x, y} when it exists and x∧ y in place of the greatestlower bound of {x, y} when it exists.

57

Definition 5 Let (P ;≤) be a non-empty ordered set.

(i) If every pair of elements x, y in P , has the least upper boundx ∨ y and the greatest lower bound x ∧ y, then (P ;≤) is called lattice.

(ii) If every subset of P has the least upper bound and the greatestlower bound, then (P ;≤) is called complete lattice.

Definition 6 Let L and K be lattices. A map ϕ : L → K is said to bea homomorphism (or lattice-homomorphism) if ϕ is join-preserving and meet-preserving, that is, for all a, b ∈ L, ϕ(a ∨ b) = ϕ(a) ∨ ϕ(b) and ϕ(a ∧ b) =ϕ(a) ∧ ϕ(b)

A bijective homomorphism is a lattice-isomorphism. If ϕ : L → K isbijective homomorphism, then the sublattice ϕ(L) of K is isomorphic to L andwe refer to ϕ as an embedding mapping.

Proposition 7 Let L and K be lattices and let ϕ : L → K. Then thefollowing statements are equivalent;

(i) ϕ is an order-isomorphism;

(ii) ϕ is order-embedding and surjective;

(iii) ϕ is an lattice-isomorphism.

Definition 8 Let L be a lattice. An element a ∈ L is said to be thegreatest element of L if a ≥ x for all x ∈ L. The greatest element of a lattice isdenoted by 1.

An element a ∈ L is said to be the least element of L if a ≤ x for allx ∈ L. The lest element of a lattice is denoted by 0.

Definition 9 Let L be a lattice with 0 and 1. An element a ∈ L is saidto be the atom of L if a covers 0. An element a ∈ L is said to be the coatom ofL if a is covered 1.

58

.

Biography

.

NAME Miss Ratana Srithus

.

DATE OF BIRTH 25 April 1979

.

INSTITUTION ATTENDED Silpakorn University, 1996-1999 :

.

Bachelor of Science (Mathematics).

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