migroup Forum VoJ. 53 (1996) 57-62 1996 Springer-Verlag New York Inc.

R E S E A R C H A R T I C L E

Modular Posers and Semigroups

J. N e g g e r s a n d H e e S ik K i m

Communicated by Boris M. Schein

Abstract Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those pc,sets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (62 +!)-free. In the case of lattices this condition means that the lattice is also Ns -free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the orginal one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.

In the s tudy of pa r t i a l ly o rdered sets var ious algebraic ob jec ts have been asso- c ia ted wi th t h e m in a var ie ty of a t t e m p t s to provide t h e m "na tu ra l " a lgebra ic s t ruc tu res which migh t then be used to advan tage in p rov id ing in te res t ing clas- sif ications of these o rdered sets. Conversely, a lgebraic objec ts have been t u r n e d into o rde red a lgebra ic ob jec ts in a m u l t i t u d e of ways in order to be t t e r ex t rac t in te res t ing a lgebra ic i n fo rma t ion by using the known proper t ies of pa r t i a l ly or- dered sets. Thus , e.g., the theory of inc idence algebras is a wel l -known example of such a pa i r ing [7, 14]. Similarly, semigroups have been equ ipped wi th orders in var ious na tu r a l ways and these have then been used to ex t rac t i n fo rma t ion abou t semigroups employ ing proper t ies of par t ia l orders in ob ta in ing resul ts on semigroups [1, 4, 5]. A less wel l -known example is the connect ion be tween B C K - a lgebras and pa r t i a l ly o rdered sets [3, 9].

In [8], J. Neggers defined a g roupo id S( . ) to be a pogroupoid if (i) x �9 y E { x , y } ; (ii) x . ( y . x ) = y - x ; (iii) (x - y) . (y . z) = ( x - y ) . z for all x , y , z e S . For a g iven p o g r o u p o i d S( . ) he defined an associa ted par t ia l o rder po(S) by x < y iff y - x = y and he then d e m o n s t r a t e d tha t po(S) is a poset. On the one hand , for a g iven pose t S ( < ) he also defined a b inary ope ra t ion on S by y �9 x = y if x < y , y �9 x = x o therwise , and proved tha t S( . ) is a pogroupoid . Thus , deno t ing this p o g r o u p o i d by pogr(S), it m a y be shown tha t pogr(po(S)) = S(.) and po(pogr(S)) = S ( < ) p rov ide a na tu ra l i somorph i sm be tween the ca tegory of pog roupo ids and the ca tegory of posets.

Now , g iven a s emig roup S define an order re la t ion -~ on 5: by se t t ing x -~ y p rov ided x r y a a d

(1) x . y = y . x =y.

Notice tha t if x - ~ y , y ~ z , t hen x = z implies x . y = y . x = y , y - z = z . y = z, con t rad ic t ing the cond i t ion tha t y # z. Also, in this case x �9 z = x - (y �9 z) = ( ~ . v ) . z = ~ . z = z a n d z . ~ = ( z . V ) . x = z . ( v . ~ ) = z . y = z , so tha t t r ans i t i v i t y of the order re la t ion follows. As usual x _ y means x = y or x -< y. Deno te the pose t ob t a ined in this m a n n e r by po*(S), i.e., we cons ider po* : S ~ po*(S) as a partial order mapping. It follows tha t i f S is a semigroup ,

NEGGERS AND KIM

then pogr(po*(S) ) -= S* is a mapping S ~ S* which associates with semigroups certain pogroupoids S*.

If S is itself bo th a semigroup and a pogroupoid then po(S) = S ( < ) , with x < y i f f y . x = y and po*(S) = S ( - ~ ) , w i t h x ~ y iff x . y = y - x = y when x ~ y. Hence let ~ mean tha t x -~ y or x = y. Then x < y iff x ~ y, whence po*(S) = po(S) and pogr(po*(S)) = S* -= pogr(po(S)) = S , i.e., the mapping S ~ S* is the ident i ty map in this case. We shall call this p roper ty of semigroups modula r* proper ty , i.e., a semigroup S is modular* iff S = S*. Fur thermore , a pogroupoid is modular* if it is also a semigroup.

Given a poset P ( < ) it is A-free if there is no full-subposet X(_<) of P ( < ) which is order isomorphic to the poser A ( < ) . If Cn denotes a chain of length n and if n denotes an ant ichain of cardinal number n , while + denotes the disjoint union of posets, then the poset (C2 + 1) (or C~ + C1 ) has Hasse-diagram

I. and may be represented as {p < q ,p o r, q o r} , where a o b denotes the relat ion of not being comparab le (i.e., a o b iff a _< b and b < a are both false). In this pape r along with several o ther results we shall demons t ra te that

T h e o r e m 1. The pogroupoid S( . ) i~ a semigroup i f f S ( . ) = pogr (P ) where P ( < ) is (C2 + l_)-free as a poser.

If L is a latt ice, then if L is non-modular as a lattice, L contains elements x < z and x V (y A z) < (x V y) A z. Thus, the la t t ice formed by {y ,x V y , y A z, (x V y) A z, x V (y A z)} is isomorphic to N5 with Hasse d iagram (see [2, 6, 12])

1

P

0

Fig. 1

where r +4 y, 1 ~ x V y, O ,-~ y A z ,q *-* (x V y) A z , p *-~ z V (y h z ) . If L ( < ) is the poset associated with the lat t ice L , then L contains a (full)subposet {p < q , p o r , q o r } with p A r and q V r corresponding to 0 and 1 in N5 as given in figure 1. Hence we obtain:

P r o p o s i t i o n 2. I f L is a non-modular lattice, then L ( < ) is not (C2 + l_)- free. �9

E x a m p l e 1. For the 16 groupoids definable on a set {p, q) the three pogroup- oids have mul t ip l ica t ion tables:

* p q * p q * p q

P P q P P q P P P q P q q q q q P q

Table 1

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NEGGERS AND KIM

cor responding to the posets {p o q}, {p < q} and {q < p} respectively. It follows tha t all three pogroupoids are semigroups, so tha t a non-modular pogroupoid must contain at least three elements. Consider now the following mul t ip l i ca t ion table:

�9 p q r

p p q r q q q r r p q r

Table 2

This is a pogroupo id cor responding to the poset {p < q,p o r ,q o r} and since q * (r * p) = q * p = q, (q * r) * p = r * p = p, it is also the case tha t this pogroupoid is no t a semigroup, i.e., it is a n o n - m o d u l a r * pogroupoid .

Thus , if we consider a poset to be modula r* iff it is (C2 + ! ) - f ree and a lat t ice L to be modu la r* iff L ( < ) is a modula r* poset, t hen we m ay res ta te several resul ts as:

T h e o r e m 1 ' . poser, and

The semigroup S is modular* i f f P = po(S) is a modular*

T h e o r e m 2 ' . I f L is a modular* lattice, then L is a modular lattice.

P r o o f . Suppose tha t a pogroupoid S is not modula r , i.e., suppose it is no t a semigroup. T h e n (p . q) - r # p . (q . r) for some choice of p, q, r (not necessari ly all d is t inct) . By def ini t ion of a pogroupoid (proper ty (i) above), (p . q ) . r and p . (q �9 r) are d is t inc t e lements of { p , q , r } . We therefore consider six possible cases, l is ted as:

(1) (p q ) . r = p , p . ( q . r ) = q ; (2) (p q ) . r = p , p . ( q - r ) = r ; (3) (p q ) . r = q , p . ( q . r ) = p ; (4) (p q ) . r = q , p . ( q . r ) = r ; (5) (p q ) . r = r , p . ( q . r ) = p ;

(6) (p q ) . r = r , p . ( q . r ) = q .

Notice t ha t in case (1), it follows tha t p # q. Similarly, (2) yields p # r ; (a) p # q; (4) q ~ r ; (5) p ~ r ; (6) q # r . Suppose tha t p . q = p . T h e n by the p roper ty (ii) (p . q) . r = (p . q) . (q . r) = p . (q . r ) , which is a con t rad ic t ion to the fact tha t p . ( q . r ) ~ ( p . q ) . r . Also p # q, since p = q implies p . q = p . Consider next the s i tua t ion p . q = q. T h e n (p . q ) . r = q . r = p (cases (1) and (2)) yields p - (q. r ) = p . p = p, whence (p. q) . r = p . (q- r) a cont radic t ion . In cases (3) and (4), (p . q ) . r = q. r = q, whence p . (q. r) = p . q = q, once more a cont radic t ion . In cases (5) an d (6), ( p . q ) . r = q . r = r , and p . ( q . r ) = p . r . Now, with (p. q) . r # p . {q. r ) , if follows tha t p . r r r , whence p . r = p. In case (6), p . (q �9 r) = q = p . r = p yields p . q = q, a cont rad ic t ion to the first case. In the last r e m a i n i n g case (5), p . ( q . r ) = p , q . r = ( p . q ) . r = r , p . r = p . ( q . r ) = p and p r r implies r < p in p o ( S ) . If q = r , t hen p = p . r = p . q = q, con t rad ic t ing the conclus ion p # q. Thus , q r r a n d p # q also, i.e., {p, q, r} conta ins three dis t inct e lements . If q < r , t hen q < p, and in tha t case p . q = p, which has been e l imina ted . Hence we conclude tha t q o r . On the one hand , if q < p, t h e n P ' q = p and if p < q, t hen r < q a n d q - r = q and p = p . ( q - r ) = p . q ,

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which is an impossibility. Hence we conclude that p o q. This demonstrates that in po(S) the (full)subposet on {p,q , r} has the form {r < p,p o q,q o r}, i.e., abstract ly it is (C2 4-1)-free, whence po(S) is not (C2 + 1)-free and therefore not modular*. Conversely, if po(S) is not (Cz + 1)-free, i.e., po(S) is not modular*, then it contains a (full)subset of the type {p < q, p o r, q o r} whence (q. r ) - p = p , q - ( r . p) = q and the multiplication in pogr(po(S)) = S is not associative. What we have shown is that :

Corollary 11 The pogroupoid S is modular* iff P = po(S) is a modular* poser. �9

The last (trivial) step is to note that a semigroup S is modular* iff it is also a pogroupoid and that a pogroupoid is modular* iff it is also a semigroup. The proof of Theorem 1 ~ is now complete. �9

To complete the circle, we may now identify those posets that are modular*, i.e., (C2 + 1)-free, by relying on some results of semigroup theory. If S is a modular* semigroup, then condition (iii) follows from condition (i) and the asso- ciative law. From condition (i) it follows that every element is idempotent, i.e., S is a band and since every subset is a subsemigroup the lattice of subsemigroups is a boolean(algebra) whence it is an ordinal sum of (singular) semigroups which are either left zero semigroups or right zero semigroups. By condition (ii) it fol- lows that i f z - y = z , t h e n x - ( y - z ) = x . y = x , y . x = y and z - ( y . z ) # y - x so that the ordinal sum is one of right zero semigroups, whose corresponding posets are antichains (see [13]). Thus, it follows that:

T h e o r e m 2. antichains.

and

S is a modular* semigroup iff po(S) is an ordinal sum of

Corollary 2. A poser P is modular* iff it is an ordinal sum of antichains.�9

If P(_<) is a modular* poser, then given x, let A(x) := {x} U {y I Y o x} be the unique maximM antichain of P ( < ) containing x. If x covers y in P ( < ) , then A(x) covers A(y) and thus if P(_<) = L (< ) , where L is a lattice, if Xl,X2 E A ( x ) , x l ~ x2, then xl Ax2 < xl and xl Ax2 < x2, whence xl A x2 e A(y ) , and therefore A(y) = {xl A x2} necessarily. Similarly, if Yl,Y2 e A (y ) , then Yl r Y2 implies A ( z ) = {Yl V y~}. Thus by an easy application of Theorem 2, which itself is a straightforward result in semigroup theory we find that:

Proposition 3. I f P = L(<) , where L is a modular* lattice and if A(x ) contains at least two elements, then A(x) covers a one element antichain and A(x ) is covered by a one element antichain. �9

R e m a r k 1. (i) This does not mean that L(<) is necessarily (2_+2)-free, where (2 + 2)-free is the ordinal sum of two antichains containing two elements. One the other hand, two such antiehains cannot cover one another in L ( < ) . (ii) A proof of Proposit ion 3 bypassing use of Theorem 2 and along lattice theoretical lines is quite feasible but considerably longer.

If P = ~ / e l Pi , is an ordinal sum of posets, then the number of linear extensions needed to obtain P as an intersection of these linear extensions does

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NEGGERS AND KIM

not exceed the max imum of the min imum number of extensions needed to obtain each P, (if this maximum exists), while it can also not be less than such a maximum. Observing that for antichains at most only two linear extensions are needed, it follows that if P is a modular poset, then no more than two extensions are needed. Hence,

Proposition 4. m o s t 2.

I f P is a m o d u l a r * pose t , t h e n P has (pose r ) d i m e n s i o n at

In [10], M. S. Rhee has characterized families in terms of exclusion results of Theorem 1 type, while in [11], I. Rival discusses N-freeness in posets, i.e., he characterizes series-parallel posets in the manner of Theorem 2. In both cases ordinal sums of antichains qualify as members of the classes defined by Rhee and Rival. One may thus suppose that there may exist binary systems corresponding to families and to series-parallel posets, for example, whose algebraic properties may be obtained and explored in a manner similar to that stated here.

A question which has not been answered completely is the following: which conditions on S guarantee that S* is modular*? Following are some examples:

E x a m p l e 2. If R is the collection of real numbers and if �9 is the usual multiplication, then p . 0 = 0 �9 p = 0 for all p and hence we obtain p _ 0 for all p , while p . 1 = 1 . p = p for all p and hence we obtain 1 _ p for all p. If p ,q ~ {0, 1}, then p . q = q . p ~ {p,q} and thus p o q in that case, and hence wc can construct the poser p o * ( R ) = R ( ~ ) . From this we can construct its associated pogroupoid p o g r ( p o * ( R ) ) = R*(*), where 1 _ p means p * 1 = p, p ~ 0 means 0 * p = 0 and otherwise p * q = q for a multiplication table:

0 1 " " P " " q

0 0 " - 0 - " 0 0 1 - P ' q

O P ' P ' q

0 q ' P ' q

Table 3

whence it follows that p o g r ( p o * ( R ) ) -= R* is also a semigroup, even though very different from R in terms of structure.

E x a m p l e 3. If (G,-) i s a g r o u p , then g . h = h . g = h means g = e , a n d t h u s the poset p o * ( G ) = G(-~) has the property that e -~ h for all h ~ e and that if e ~ {g, h}, then g o h. We obtain a "multiplication table" for the pogroupoid a * = p o g r ( p o * ( a ) ) :

* e ' - . g ' " h

e e . . . g . . , h

g g ' " g ' " h

h h ' " g ' " h

Table 4

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whence it follows that G* is again a semigroup, i.e., once more we have an affirmative answer to the question, even though the structure of G* is very different from tha t of G.

Given the possible complexities in semigroups it is not clear that if S is a semigroup then S* is a modular* semigroup. On the one hand it makes for a good conjecture. On the other hand, should it fail to be so, the separation of semigroups into two classes on this basis would be possibly an interesting one.

A c k n o w l e d g e m e n t s : The authors are deeply grateful to the referee for the valuable suggestions.

R e f e r e n c e s

[1] W. D. Burgess and R. Raphael, On Conrad's partial order relation on semiprime rings and semigroups, Semigroup Forum 16 (1978) 133-140.

[2] G. Grs " General lattice theory," Academic Press, New York, 1978.

[3] C .S . Hoo, MV and BCK-algebras on posers, Math. Japonica 36 (1991) 137- 141.

[4] C . J . Johnson, Jr. and F. R. McMorris, Arbian's order relation for semigroups, Semigroup Forum 16 (1978) 147-151.

[5] N. Kehayopulu, On regular le-semigroups, Semigroup Forum 49 (1994) 267- 269.

[6] D. Kelly and I. Rival, Planar lattices, Canad. J. Math. 27 (1975) 636-665.

[7] P. Leroux and J. Saraill6, Structure of incidence algebras of graphs, Comm. in Algebra 9 (1981) 1479-1517.

[8] J. Neggers, Partially ordered sets and groupoids, Kyungpook Math. J. 16 (1976) 7 - 20.

[9] J . G . Raftery and T. Sturm, On direct products of BCK-chains, Math. Japon- ica 36 (1991) 27-37.

[10] M. S. Rhee, Some properties and applications of P-graphs and families to posers, Diss., Univ. of Alabama, 1987.

[11] I. Rival, Stories about order and the letter N , Contemporary Math. 57 (1986) 263-285.

[12] V. N. Salii, " Lattices with unique complements," Tran. Math. Monographs Vol. 69, Amer. Math. Soc., 1988.

[13] L. N. Shevrin and A. J. Ovsyannikov, Semigroups and their subsemigroup lattices, Semigroup Forum 27 (1983) 1-154.

[14] E. Spiegel, Radicals of incidence algebras, Comm. in Algebra 22 (1994) 139- 149.

Dept of Mathematics University of Alabama Tuscaloosa, AL 35487-0350 U.S.A. Email: jneggers@gp.as .ua. edu

Received August 22,1994 and in final form December 22, 1995

Dept. of Mathematics Education Chungbuk National University Chongju, 360-763, Korea Email: heek•162 r ac . k r

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