Commentationes Mathematicae Universitatis Carolinae
William H. Cornish A ternary variety generated by lattices
Commentationes Mathematicae Universitatis Carolinae, Vol. 22 (1981), No. 4, 773--784
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This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE
22.4 (1981)
A TERNARY VARIETY GENERATED BY LATTICES William H. CORNISH
Abatract: The class of all subreducts of lattices, with respect to the ternary lattice-polynomial sCxfyfz) * * xACyvz), is a 4-baeed variety. Within ita lattice ot aubvarietiea, the subvariety of distributive supremum algeb ras ia atomic and needs 4 variables in any equatlonal des cription. Key words: Subreduct, lattice, neariattice, distribu— tivity. Classification: 03005, 06A12, 06D99, 08B10
0* Introduction. Consider the variety of algebras (A;a) sf typ#<3> that satisfy the following identities
CS1) aCxfxfy) » xf
CS2) aCxfy,y) »s(yfxfx)f
CS3) s(xfyfz) «s(xfzfy)f
(54) wAa(xfyfz) *s(iAx,y,zl,
(55) 8(xfa(yfvfw)f aCzfvfw)) £aCxfvfw), where A is the derived operation given by
CD1) xAy * aCxfyfy), and £ Is the derived relation defined by CDS) x^y if and only if x « XAy. Then, we ahow that an algebra la in thla variety if and only if it ia a subalgebra of s reduct (L;a), where CL; A ,V ) ia
- 773 - a lattice and s(x,y,z) » xACyvz). The variety generated a single such lattice-reduct (L;s) has already been atudied by Baker £13; he showed that the variety is congruences- distributive but not 3-distributive« Of course, his result extends to our larger variety. However, this can be obtain ed from Hickman £21, as each reduct (A;j) of an algebra (A;s), where j is the derived ternary operation given by CD3) jCx,y,z) » sCy,XAy,yAz), is a join algebra. We demonstrate this by showing that each algebra is a nearlattice wherein :j(x,y,z) » CxAy)v(yAz). Baker's result has already been exploited by Berman 12}• In the presence of CS1) - (S4), (S5) is implied by
CS6) W!AS(x,y,z) sr s(x,WAy,WAz). The subvariety defined by the five identitiea (Si) - (S4) and (S6), consi3ts of subalgebras of reducts of distributive lat
tices and so is the variety generated by (2;s), where (2; A fv) is the two element lattice. Thia variety waa described by Lyndon C8; Theorem 33 in 1951, uaing nine identities.
-«• Subreducts. Before proceeding to the main results, we would like to make some remarks on subreduct9. that i3 subalgebras of reducts, of the algebras in a variety. Let V be a variety of finitary algebras, p be an n-ary ^-polynomial, and V be the class of all subalgebraa of al gebras having the form C.A;p), where k is the underlying set of a J-algebra. Then, Jp is a class of algebras of type
- 774 - groups and the cancellation law can fail in homomorphic i- mages of such semigroups. In general, V- is closed under the formation of products and subalgebras. A sufficient condition to ensure closure under homomorphic images is as follows. Suppose p is congruence-determining in the sense that the V-congruences on a ^-algebra are precisely the V -cong ruences on its V -reduct. Also, assume that JJ enjoys the Congruence Extension Property. Then, it is immediate that Y is closed under homomorphic images and so is a variety. In the above case concerning Abelian groups, p is congruen ce-determining but the Congruence Extension Property fails for commutative cancellafcive semigroups. An instance where these conditions hold is supplied by the case of Y being the variety of Boolean algebras (A; A , V , ' ,0,1) and p being the binary polynomial pCx,y) * x*y » xAy . Here, Yp is the variety of implicative BCK-algebras; for an account, see [3] and especially Corollary 1.5, therein. Thus, Y has the Con gruence Extension Property; in fact, any variety of BCK-al gebras has the Congruence Extension Property, [5; Theorem 3.13 • It is not hard to show that p(x,y) = x*y is congruen ce-determining on a Boolean algebra, or alternatively, this can be deduced from t4, Corollary 3.131. However, each of these two conditions is not necessary for V to be a variety. For instance, when Y is the variety of (distributive) lattices U;A,V) and p is given by pCx,y) » XAy, Y is the variety of semilattices; here, V has the Congruence Extension Property but p is not congruence
- 775 -determining. On the other hand, the main result of this pa per | as described in Section 0, supplies an example where the Congruence .extension Property fails (because it fails in any variety of non-distributive lattices) and p, that is s, is congruence-determining. Actually, Baker commented on this property of s in tl; Note 1, p. 1433; this can also be seen from Hickman C7» Proposition 2.23, since (D3) of Secti on 0 gives Hickman's j as a polynomial in terms of s. Anot her such example is provided by Hickman's join algebras, see 17; Proposition l.l(iii), Theorem 2.13. Thus, the main res ult of this note has a significance with respect to a model* theoretie problem*
2* Main results. A partially ordered set is said to ha ve the upper bound property if any two elements possess a su pra mum f whenever they have a common upper bound. A near latti ce is a lower semllattice with the upper bound property. In any nearlattice (A;A,£), j(xfyfz) * (xAy)vCyAz) is defin ed on the whole of A. The resulting ternary algebras (A;j) are equationally definable and are called .join algebras. Ac tually, in £7; Theorem 2.1, Proposition 2.43, Hickman showed that the variety of join algebras and their ho isomorphisms is isomorphic to the category of nearlattices and their near ia ttice-homomor phi sms; the isomorphism commutes with the for getful functors to the category of sets. We will give no fur ther details; additional information on neariattices and their congruences etc. can be found in £4; Section 3]. It should be mentioned, however, that a lower semllattice is a
- 776 - nearlattice if and only if the new semilattice, that results from the addition of a largest element, is really a lattice.
Hence, the variety of join algebraa i8 the variety Vpf when X is the variety of latticea and p i8 j. We make use of the notation of Section 0. A ternary al gebra (A;s), satiafying (SIJ - (S5), will be called a suort- mum algebra.
Lemma 2.1. L&1 (A;s) fee a, a\jpremum algebra. Then,
(A;A ,£) jla fl ng§rlqttl9fl. for ,aay a,b,ceAf a(a,bfc) «
* a A (b v c) f jhaa b v c exiata. Proof. By (Sll and (Dl), aA a = s(a,a,a) 3 a. iron (S2), aAb » bAa. From (S4) f a A (bA c) * aA s(b,c,c) » s(a A b,c,c) =
= (aAb) AC. Thus, (D2) says that (A;A9-&) la a lower semi- lattice, wherein £ la the induced partial order. Suppose b,c,£a and let d « s(a,b,c). Due to (S4) and
(Sl)f bA d »aCbAafbfc) » 8(b,bfc) * bf i.e. b£d. Similar- ly, c£df and d la a common upper bound of b and c. Let e be another such bound. Then, d ». s(a,b,c) = s(a,bAe,cAe) =
= s(a,s(b,e,e), s(c,e,e))^ s(a,e,e) -*aAe£ef by (S5). Hen ce, d » bvc. When bvc exists, the above reasoning shows that bvc -
* s(bvc,b,c). Hence, aA(bvc) a aAavbvcfbfc) = s(aA(bvc,), b,c) » (b vc)< As(a,b,c)& s(a,b,c), due to two applications of
CS4). But due to ($51, alafbfe) » s(a,b/s (bv c) ,c/s(bv c)) »
* s(a,sCb,bv cfbvc)f 8(c,bvc,bvc))^s(a,bvc,bYc! »
=-aA(bvc). Hence9 s(a,b,c) =-aA(bvc). Notice that the lemma implies that j, aa defined by (D3), is given by j(xtyf*l » (xAy)v (JTAZ) on the underlying r»ay-
- 777 - lattice of a supremum algebra. Hence, the reduct (A;j) of a supremum algebra (A;s) is a join algebra, and so, by Hick man 173, we obtain
Corollary 2.2. The variety of supremum algebras is con gruences-distributive - but not conffruence-3-diatributive. In order to establish the characterization of supremum algebras as subreducts, we need to introduce the appropriate ideal-theoretic concepts. An s-ideal of a supremum algebra (A;3) is a non-empty subset K such that, for any aeA and k-^k^e K, s^(a,k-j, ,k2) e Kf When a^k and keK, a - s(a,k,k), and so a£ K. If kj^k^eK and k^v k^ exists, then Lemma 2.1 implies k,v k« = s(k, v k^,k, jk^) and so k,v k^e K. Thus, an s-ideal is a nearlattice-ideal of the underlying nearlattice. Nevertheless, the true nature of s-ideals is a mystery to us, even though we can exploit them* When K, and Kp are s-ideals of a supremum algebra (A;s), there are k-^e K and k2^ K and so, k^A k^ =* s(k^A k^k^,]^) = -» s(k-rA k^,^,^), says that k^A k^ is in the set-intersec tion K-J-A Kp. It follows that K^AK*, is an s-ideal. Also, let
TQ = A a e A: a =s(a,k1,k2), k^cK, k^e iy and T ^ » i a € A: eT for n 1# Tnen we naVe Ja « s(a,tlft^), tx'*2 n^ ~ > inducti vely defined the aequence K^, IL& TQ£ T^s ... £ Tn^T ^S ... . Moreover, it i9 not hard to show that U\T :n>OJ is the smalleat s-ideal containing both K, and K^. Hence, when or dered by set-inclusion, the s-ideals of (A;s) form a lattice, wherein the infimum and supremum of s-ideals K, and K^ are given by K-,0 K^ and KjV K^ =- U-CTn:n>0{, respectively. Also, it is not hard to see that for any beA, Cb3 a{aeA.a^b}
778 - is the s-ideal generated by b. In general, the s-ideal gene rated by a non-empty subset B of k is U{Sn-n.£0?f where SQ *
*-CaeA:a * 8(a,blfb25) fb^$b^€ B}f S^ =--£ae&-a -» s(afrlfr2) f rlfr?eSnif for n£l.
Lemma 2.3. Let (A;s) be a supremum algebra anda.b.ccA. Then, in the lattice of a-ldeals (b3v(c3 *-£d€A:d «
- a(dfbfc)lf and consequently (a3n ((b3 v (c3) ** (s(a,b,c)].
Proof. Let T -*-£d£A:d -=s(dfbfc)f. Then, bfceTf by
(SI) and (S3). Also, if d-^ d and d = s(dfb,c) then d-^ =-
=- (djLfbfc)f due to (S4). In other words, T is hereditary.
Now, if t^y T and «£Af then, because of (S5) and (S4), 3 sCe,^-^) s(e,s(tlfb,c), s( t2,b,c)) «£ s(efbf c) f and s(e,b,c) = s(e,b,c)A s(e,b,c) » s(s(e,b,C)A e,b,c) *
-* s(s(efb,c),b,c). Hence, s(e,b,c)cT, and so s(eftift.£)€ Tf too. Thus, T is an s-ideal containing b and c. It immediately follows that T = (b3vCc3. The remaining assertion follows quickly from (S4>, and its consequence that s(a,b,c)^ a.
Theorem 2.4. Lg£ L be the variety of all lattices and a fre the ternary lattice-polynomial s(xfyfz) = xA(yvz).
Then. Ls fa, frfte yylrty Pf mprtWM ftlffit-!M« Proof. It i8 easily verified that each algebra in L satisfies (SI) - (S5). On the other hand, a supremum algebra
(A;sX is in L f as the map a—*> (a3 is a supremum algebra-em bedding of (A;s) into the s-reduct of its lattice of s-ideals. We now turn to distributivity. A nearlattice is dflia|rl- butive. when the infimum distributes over existent finite su-
- 779 pre ma. This is equivalent to the associated join algebra2 sa tisfying the identity wAj(x,y,:) « J(WA x,y,wAz), where xAy » j(x,y,x); see £?; Theorem 3.31. More importantly, a nearlattlce is distributive if and only if either each ini tial segment is a distributive sublattice or the lattice of nearlattice-ideals is distributive or the finitely generat ed nearlattice-ideals form a distributive lattice, when or dered by set-inclusion. The equivalence for these last two conditions is contained in the proof of L'6; Theorem 1.2].
Theorem 2#5. The following conditions pn a ternary al- gebrq (A;s) are equivalent* (i> (A;s) IS a supremum algebra satisfying the identity (S7) s(x,y,z) =- S(X,XA y,x/\ z).
(ill (A;s) IS a supremum algebraf whose lattice of s- jdeals la dlgtrifrutlye. (iiil (A;s) satisfies the identities (SI) - (S4) and (SS^t where A is defined by (Dl). Proof, (i) «-.• (ii). It follows from (i) and Lemma 2.1 that each initial segment of the underlying nearlattice is a distributive lattice. It also follows from (i) and Lemma 2.1 that each nearlattice-ideal is an s-ideal. Thus, the two concepts of ideal coincide here. Aa the nearlattice is dis tributive, (ii) follows. (iil*--> (iii) holds because a distributive lattice sa tisfies (S6).
(iii; ~-Mi). Reasoning along much the same lines em In the proof of Lemma 2.1, we see that (ill) implies that
(A;A>^) is a nearlattice* Also, s(a,btc) = bvc, whenever a is
7Ö0 - an upper bound of both b and c. Due to CS6), sCx,y,z) * ss aCxAX,y,z) = xAs(x,y,z) =- SCX,XA y,xA z). Hence, (S7) holds and s(x,y,z) = (XA y) v CXA z). Then, CS6) says that the infimum distributes over such a supremum. Using these observations, it is possible to express the left side of (S3) as a supremum of infima, and so establish (S5).
Corollary 2.6. Let J) be the variety pf distributive lattices and s be the ternary D-Polynomlal aCx,y,z) -»xACyvz)=
-= CxAy)v(xAz). Then. BQ jg the variety pf a,!), qjlgebraa satisfying the conditions of Theorem 2.5. Call the algebras of Theorem 2.5. distributive supremum algebras. The above results show that Hickman's distributive? join algebras and distributive supremum algebras are defini- tionally equivalent. And, perhaps, it would be more natural to describe join algebras by CxAy)v/CxAz) instead of j. Co rollary 2.6 ensures our remarks at the end of Section 0. Hickman's equational base for the variety of distributive su premum algebras has 9 identities, as does Lyndon's. Ours has 5 identitittes. However, both the variety of supremum algebras, and its subvariety of distributive algebras, can be defined by using at most 4 identities. This is because of Corollary 2.2, and Padmanabhan and Quackenbush L9; Theorem 13. A related question is that of the minimum number of va riables needed in an equational base for one of these varie ties.
Theorem 2.7. An eotiational base for either supremum al- fi?frrag or distributive gupremytfl aMefrrfl? need a,j jLeggt? 4 l&z
- 781 irJj&lsj.* And 4 ig sufficient for djUtfrfrfriUYg amxiww gte
Proof. Suppose 3 variables are sufficient for either of the varieties. Because of Theorem 2.5 and (S7), 3 vari ables suffice for distributive supremum algebras. Hence 3 variables suffice to equationally describe distributive join algebras. Now consider the 5-element modular non-dis tributive lattice. The associated join algebra is not dis tributive, yet all of its 3~generated join subalgebras are distributive. This gives the desired contradiction. The re maining assertion follows from Theorem 2.5(iii). When looking for examples of supremum algebras, it is important to observe that any hereditary subset of a latti ce is closed under s, and so becomes a supremum subalgebra. We do not know whether all supremum algebras arise this way. But distributive supremum algebras doI As observed in the proof of Theorem 2.5, the s-ideals and the nearlattice-ide- als coincide on a distributive supremum algebra. Moreover, the finitely generated nearlattice-ideals form a lattice when the near lattice is distributive, c.f. t6; Theorem 1.23. Then, due to the upper bound property, a distributive sup remum algebra Is a hereditary subset of its lattice of fi nitely generated s-ideals. Another feature of distributive supremum algebras is that they are s-isomorphic if and only if they are order-i- somorphic. In general, this is not the case. For let us con sider the 4-element nearlattice ., \fhich has 3 atoms a, b, and c, and smallest element 0. It is a hereditary subset of
. - 782 the following 4 lattices L^-I^L^-L^. Here, L^ ia the Boo lean lattice with a,b, and c as atoms; the associated al gebra AJL » (A;s) is distributive, but not subdirectly irre ducible. Then, L^ is the 5-element modular non-distributive lattice; the algebra A^ - (A;s) is simple and not distribu tive. The lattice L-* haa 5 elements; the new elements are d and e, d =- bvc, e = avd =avbvc; the algebra A-j = (A;3) is subdirectly irreducible, but not simple. The fourth lat tice L, has 6 elements; the new elements are d,e,f, d =- avb, e » bvc, f =• dve; the algebra A* a (A;s) is also subdirect ly irreducible but not simple. Up to s-isomorphism, A-* - A. are the only distinct supremum algebras which contain A as a
subnearlattice. Because of congruence-distributivity, A2» An and A* generate distinct varieties which cover the variety of distributive supremum algebras. Thus, to us, the study of the lattice of subvarietiea of supremum algebras seems hope lessly difficult.
References Lll K.A. BAKER: Congruence-distributive polynomial reducts of lattices, Algebra Univ. 9(1979), 142-145. L21 J. BERMAN: A proof of Lyndon's finite basis theorem, Discrete Math. 29(1980), 229-233. L31 W.H. CORNISH: A multiplier approach to implicative BCK- algebras, Math. Seminar Notes Kobe Univ. 8(1980), 157-169. [41 W.H. CORNISH: 3-permutability and quasi commutative BCK- algebras, Math. Japonica 25(1980), 477-496. L51 <..H. CORNISH: On IsSki's BCK-algebras, to appear as Chap ter 9 in Algebraic Structures and Applications,
783 Marcel-Dekker, 1981. 16} W.H. CORNISH and R.C. HICKMAN: Weakly distributive se- milattices, Acta Sci. Math. Hungar. 32(1978), 5-16. t71 R# HICKMAN: Join algebras, Commun. in Alg. 8(1980), 1653-1685. t8l R.C. LYNDON: Identities in two-valued calculi, Trans. Amer. Math. Soc. 71(1951), 457-465. L9J R. PADMANABHAN and E.W. QUACKENBUSH: Equational theo ries of algebras with distributive congruen ces, Proc. Amer. Math. Soc. 41<1973), 373-377.
School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park, 5042, South Australia, Australia. (Oblátům 2.6. 1981),
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