DEMONSTRATIO MATHEMATICA Vol. XXV No 4 1992

Radoslav M. Godowski

BISEMILATTICES WITH CO-ABSORPTION LAW

1. Introduction

An (L,+,-) with two binary operations + and • is said to be a bisemilattice if both of its reducts (L,+) and (L,-) are , i.e. both operations + and • are idempotent, commutative and associative. A structure (L,+) yields a partial order on the L by setting x^+y iff x+y=y. Similarly, we define xs'y iff x-y=y. Therefore, we can define two (in general different) partial orders on L. The two partial orders and s' on L coincide iff the operations + and • coincide on L. Since in this case the bisemilattice is a set with a operation considered twice, once as meet and join, we will simply call it a semilattice (for details see [2]). We use the following notations and conventions. Meets x-y will simply be denoted xy. In the figures, the left hand diagram always represents the reduct (L,•) with the partial order s"; the right hand diagram represents the reduct (L,+) with the partial order s . Finally, if X is a class of bisemilattices, then by [X] we denote the smallest variety containing X.

2. Basic definitions and notions

First, we recall some identities and definitions from [2]: (C) (x+y)'y = x+y (co-absorption law) (1) x+y+xy = x+y (3) (xy+y)•(xy+x) = x+y. We denote by BC. (BC1,BC13) the variety of bisemilattices satisfying (C) ((C) and (1), (C), (1) and (3) respectively). 948 R.M. Godowski

Note that the variety SL of semilattices is the least nontrivial subvariety of BC. In [2] it is given an example of a bisemilattice from BC13 which is not a semilattice (we denote it by C^)• The C^ plays an important role in this paper. The diagram of C_ is the following:

o 1

Now we give an equivalent condition to replace the identity (C) for bisemilattices.

Proposition 1. Let (L,+,*) be a bisemilattice. Then (L,+,•)eBC iff the partial order s" is an extension of i.e. the following conditions holds:

(C') xs+y =—> xs'y.

The proof is obvious.

Observe that any of the above identities is an identity of two variables. Thus a bisemilattice (L,+,-) belongs to the class BC (BC1, BC13) whenever any its two-generated subalgebra belongs to this class. A two-generated bisemilattice that belongs to BC can be, in general, infinite (for example CY^ in [3]). But if generators are comparable then the situation is much simpler.

Lemma 1. Let (L, + ,-)<=BC be generated by a two element set {u,v} with ui'v. Then either L is a two-element semilattice or L is isomorphic to the C2. Moreover L is a semilattice iff us+v. Both possibilities are ilustrated below. Bisemilattices with co-absorption law 949

u+v o v o o v v o ,o u+v u o o u u o' O V U O U+V = U* V =v

Proof. The condition ui'v means that u»v=v. Using (C) we get v^+u, i.e. u+v^u. If us+v, i.e. if u+v=v, then, of course, L is a two-element semilattice. If not, i.e. if u+v/v, then vi+u+v. Using (C) we get uî'vî'u+v. Therefore L is isomorphic to the C2.

Using the above lemma we can prove that the C2 is a spitting algebra in the variety BC. Theorem 1. Let (L,+,*)€BC* Then either L is a semilattice or C2 is a subalgebra of L. Proof. Suppose that LeBC and L is not a semilattice, i.e. the two partial orders on L do not coincide. From (C) follows that there exists a,beL such that a^'b, but a?i+b. It follows from the above lemma that {a,b,a+b} forms a subalgebra of L isomorphic to the C2. Now the of all varieties of bisemilattices with co-absorption law can be presented as in the figure below:

o BC

o SL

O T

3. Congruences in the variety BC Recall that an equivalence relation p on L is a congruence on a bisemilattice (L,+,-) iff for any x,y,zeL the fol- lowing conditions hold: 950 R.M. Godowski

(Rl) x p y xz p yz, (R2) x p y =» x+z p y+z.

Definition 1. Let (L,+,*)eBC. A filter on L is a nonempty subset F of L such that: (aeF, as'b) » beF for any a,beL.

Examples. Let K be a subset of L. 1) If the set [K) = {xeL: ys'x for any yeK} is nonempty, then it is a filter. In particulary, if K is an one- -element set, K={a>, then this filter is denoted by [a). We call it a principal filter. Note that the filter [K) is an intersection of a family of principal filters, namely:

[K) = Pi [a) • aeK 2) If the set

Proposition 2. The relation Rp generated by a filter F, defined as above, is a congruence relation on the bisemilat- tice L.

Proof. It is easy that R„ is an equivalence relation. We r show that the conditions (Rl) and (R2) are satisfied. If x=y, then the situation is clear. Suppose that x,yeF. Since xs'xz and y^'yz we infer that xz,yzeF. Moreover x^+x+z and y-+y+z. Using (C' ) we obtain xs"x+z and y^"y+z. Thus x+zR„y+zr , what completes the proof. Remark that in (L,+,«)eBC the following conditions are equivalent:

(1+) x~+1 for any xeL,

(1•) xs'l for any xeL. Thus, if there exists the greatest element relative to one Bisemilattices with co-absorption law 951

order, then it is the greatest one relative to second order.

Proposition 3. Let (L,+,-)eBC and let l€L be the greatest, join-irreducible element of L, i.e. for any x,yeL the following hold:

(1) (2) x+y=l =» [x=l or y=l]. Then the relation xpy s [x=y=l or x,y*l] is a congruence on (L,+,•)•

Proof. The fact that p is an equivalence relation is clear. If x=y=l or z=l, then, of course, (Rl) and (R2) hold. Suppose that x,y,z*l. Applying (2) we get x+z,y+z*l. From the condition (C' ) it follows that xz^'x+z and yz^'y+z. Thus xz,yz*l. Therefore (Rl) and (R2) hold.

4. Subdirectly irreducible in the variety BC

It is clear that there exists only one two-element subdirectly irreducible algebra in BC, namely two-element semilattice. Now we give some necessary and some sufficient conditions for a bisemilattice from BC to be subdirectly irreducible algebra.

Theorem 2. Let (L,+,-) be a subdirectly irreducible algebra from BC of cardinality greater then 2. Then there exist three distinct elements l,p,aeL such that 1 is the greatest element of L, p is the unique coatom with respect to s" in L and 1 is the join of p and a, i.e. for any xeL:

(1) xs+i, (2) x=l or xs'p, (3) a+p=l. This situation can be ilustrated as follows: 952 R.M. Godowski

Proof. Let us consider a family {R_cl: aeL} of distinct congruences, where R denotes the relation generated by a Si principal filter [a). Observe that the intersection of this family is the trivial relation. Because L is subdirectly irreducible, then R 3 must be trivial for some aeL. But R a is trivial iff xs'a for any xeL, i.e. iff a is the greatest element of L. Now, for K=L\{1} the intersection of the family {Ra : aeK} must be nontrivial, because Rc l is nontrivial for any a*l. But this intersection is equal to the relation Rr„. generated by the filter [K) . It is easy to prove that in this case, [K) is a two-element set {l,p} and p is the unique coatom with respect to s" in L. Finally, the above condition (3) follows directly from Proposition 3. Observe that the smallest nontrivial congruence on the above bisemilattice (L,+,*) is equal to R . P Definition 3. Let (L,+,l) be a semilattice with the greatest element 1 relative to order s . We say that the L is codisjunctive iff

(*) for any a,beL such that as+b there exists ceL such that

as+c<+l and b+c=l. Note that (L,+,>) is codisjunctive iff there exists a (B,+,*,',0,1) such that L is a co-dense subset of B, i.e.: 1) if beB, b*l, then there exists aeL, a*l such that b^a, 2) the partial order on B is an extension of the s . Proposition 4. Let (L,+,»)eBC. Suppose that (L,+,l) is codisjunctive and that the conditions (l)-(3) of Theorem 2 are satisfied. Then (L,+,*) is subdirectly irreducible. The proof is obvious.

Examples. 1) The bisemilattice presented below satisfies the conditions (1)-(3) of Theorem 2, but it is not subdirectly irreducible. Bisemilattices with co-absorption law 953

0 1 1 ° 1 | p o o a 0 a | 1 o 0 o 0

2) This bisemilattice is subdirectly irreducible although it is not codisjunctive. 0 1 1 o 1 op | a o op 0 a | 1 0 00 5. The class BCCh of bisemilattices

Let (L,+) be a join-semilattice. Consider the family of such bisemilattices from BC that their join-reduct is equal to

(L,+). This family is non-empty (e.g. <"=<+) and can be ordered by a natural partial order in the following way:

L1^L2 s <* on Lj is an extension of s' on L^. Observe that the maximal elements in consideration of the above partial order are bisemilattices in which the meet- -reduct (L,*) forms a chain.

Definition 6. The BCCh is a class of bisemilattices from BC in which the reduct (L, •) forms a chain.

Proposition 7. Any bisemilattice from the class BCCh belongs to the variety BC13. The proof is straightforward. Observe that the class BCCh is closed under taking of homomorphic images, subalgebras and ultraproducts, but it is not closed under taking of products of algebras. The natural example of bisemilattices from BCCh are algebras N2(n) and N2(w) introduced in [1]. We recall here definitions of these algebras.

Definition 7. Let N be the set of all positive integers. For x,yeN we shall denote by [x,y] their least common 954 R.M. Godowski multiple. The symbol D(n) stands for the set of all divisors of neN. Now we define the following algebras:

N2(w) = < N,[x,y],max(x,y) >

N2(n) = < D(n),[x,y],max(x,y) >.

The bisemilattice N2(n) is a subalgebra of N2(w) for any neN.

Moreover N2(n) is a semilattice iff n is a prime power (see [ID- AS a consequence of Proposition 4 we obtain some suf-

ficient condition for N2(n) to be subdirectly irreducible: 2 Corollary. Let n*2 be a natural number such that p is not a divisor of n for any prime number p. Then the bisemilattice

N2(n) is subdirectly irreducible. Examples:

1) The algebra N2(20) is not subdirectly irreducible. The equivalence relation identyfying 5 and 10 only is also a minimal nontrivial congruence relation. 2) The algebra N_(18) is subdirectly irreducible although 2 . . . 3 is a divisor of 18.

REFERENCES

[1] J. Dudek: On bisemilattices III, to appear. [2] J. Dudek, A. Romanowska: Bisemilattices with four es- sentially binary polynomials, Preprint nr 569 (XII 1980), Technische Hochschule Darmstadt. [3] R.M. Godowski: Varieties of bisemilattices with co- -absorption law, to appear.

INSTITUTE OF MATHEMATICS, WARSAW UNIVERSITY OF TECHNOLOGY, 00-661 WARSZAWA

Received November 29, 1990.