Rachůnek, Jiří

English

Rachůnek Jiří

University of Szeged

Acta Sci. Math., 54 (1990), 241—246 A characterization of ^-distributive semilattices J. R A C H O N E K The notion of a distributive ordered set which generalizes the notion of a dis-tributive lattice is introduced in [3], where there are shown some properties of such ordered sets. In [2] there are described ordered sets having a similar importance for distributive ordered sets as the pentagon and the diamond have for distributive lattices, i.e. on certain conditions they are not included in a distributive ordered set (e.g. as its strong subset) and each non-distributive ordered set contains at least one of those sets as an L[/-subset. (For the definitions of an LCI-subset and a strong subset see below.) The aim of this paper is to describe the semilattices which are distributive ordered sets. Let A=(A, be an ordered set. If BQA, then we denote LA(B) = {x£A\ x^b, for all b£B), UA(B) = {y£A; y^b, for all b£B}. If it is not a danger of misunderstanding, we write also L(B) and U(B) instead of LA(B) and UA(B). For B={a1, ...,«„} we use also the forms L(B)=L(a1, ...,a„) and U(B) = U(ai, ...,an). D e f i n i t i o n 1. An ordered set A is called distributive if t L(U(L(a, c), L(b, c))) = L(U{a, b), c) for all a, b, c£A. R e m a r k 1. It is clear that in any ordered set A it holds L(U(L(a, c), L(b, c)))g QL(U(a,b),c) for all a,b,c£A. Hence for the distributivity of an ordered set it suffices to verify only the identity with the opposite inclusion. R e m a r k 2. A lattice A is distributive if and only if it is a distributive ordered set. (See [3].) Received May 20, 1988 and in revised form June 27, 1989. 242 J. Rachunek Recall that a semilattice A=(A, V) is called distributive (see [1, p. 135]) if for any a, b, x£A it holds the following condition: If x^aVb, then there exist a l 3 b^A, ^ S a , b^b such that x=a1Vb1. To distinguish two notions of distributivity, a semilattice which is simulta-neously a distributive ordered set will be called an ¿»-distributive semilattice. We will show a connection between these notions. P ropos i t i on 1. Every distributive semilattice is o-distributive. Proo f . If A=(A, V) is a semilattice, a, b, c£A, then L(U(a, b), c)—L(aMb, c). Let A be a distributive semilattice, a, b, c, x£A, xSc, x=a\Jb. Then there exist a1,b1£A, a^a, b^b such that x=ax\Jb1. Let y£ U(L{a, c), L(b, c)). Then a^y, b1^y, hence x=a1Vb1Sy, and therefore L(aVb, c)QL(U(L(a, c), L(b,c))). R e m a r k 3. The converse implication is not true. For example, the semi-lattice A={a, b, c}, where a<c , b<c (see Fig. 1), is o-distributive but it is not distributive. D e f i n i t i o n 2. a) A subset M of an ordered set A is said to be an LU-subset of A, if for each a, b£M: (i) LM(a, b)-0 if and only if LA (a, b)=0; (ii) U„(a, b)=0 if and only if UA(a, b)=0. b) A subsemilattice M of a semilattice A=(A, V) which is an LU-subset of A (i.e. M satisfies the condition (i)) is called an JL£/-subsemilattice of A. Theorem 2. Let a semilattice A—(A, V) do not be o-distributive. Then it contains an LU-subsemilattice isomorphic to one of the ordered sets M2, A/4, N3, N4. c Fig. 1 (See Fig. 2.) d d d d M2 r M 4 a c a b c N3 •a e N4 c Fig. 2 o-distributive semilattices 243 Proof . If a semilattice A is not ¿»-distributive, then there exist a,b,c£A such that L(U(L(a, c), L(b, c))) c L(aVb, c). I. Let a<c . Then L(U(L(a,c), L(b, c)))=L(U(a, L(b, c))), and thus L(U(a, L(b, c)))cL(aVb, c). Clearly a\\b, b\\c. (a) Firstly let us suppose L(b,c)=0. Then there exists x£L(aVb,c) such that x ^ a . (a) Let x>a. Then aVb=bVx, a\!b>b, b\\x. From that we also have aMb^-x. Therefore the set Tx = {a, b, x, aNb} is a subsemilattice of A. Furthermore L(a, b)Q ^L(b, x)QL(b, c)= 0 , hence Tx is an LEZ-subsemilattice of A isomorphic to M2 . (P) Let x\\a. Let us denote T2={a,b, a\!x, aSJb). We have aVx^aVb and flcaVx. Furthermore aMb^c. In the case c~za\lb, we obtain a\!bs ^flVx, in the case c||aV6, we have aVx<c, a\!x<a\ib. Therefore it always holds a\lx<a\lb. In addition, we have b<a\lb. Let us show that bj|aVx. In fact, if aVxSb, then a contradiction, and if b-^aVx, then aNb-^aVx, a contradiction, too. Therefore T2 is a subsemilattice of A, and because L(a,b)QL(b, aVx)Q £L(b , c)= 0 , T2 is an Z,£/-subsemilattice of A isomorphic to M2. (b) Let now L(b, c)^ 0 and let v£L(b, c). Since L(U(a, L(b, c)))cL(aV6, c), there exist x£L(aVb, c), y£U(a, L(b, c)) such that x^y. (a) Let x>y. Let us denote T3={b,x,y,v, aVb). Then from a < x we obtain aVb^xVb, and since evidently xMb^aSIb, we have yVb=a\Jb. Further it is clear that v<b and v<y. Since c\\b, we have x<aVb. If ¿ S x , then b^-a, and if b^x, then x=a\/b, hence it must hold Z>[|x. Analogously we can prove b\\y. But this means that T3 is an Li/-subsemilattice of A isomorphic to Mi. (P) Let x|| j . Let us denote Tt = {b, a\Jv, xVaVv, v, aVb}. Since v<b, x-^aNb and a^b, we have xVaNvSaNb. Let us suppose x\!a\!v=cNb. Then xMaNv^b, hence cVxVaVv^bVc. But cVxVaVv=c, therefore csfe, a contradiction. Thus it must be xVa\tv-<.a\fb. Since x\\y, we obtain xl£a\/v, hence xVaMv^aMv, and so a\lv<x\la\/v. Further it is evident that v<aVv, v<b, b<aVb. At the same time, if bsaVv, then b^a, and if b^aMv, then f>Sc, a contradiction. Thus b\\a\/v. Similarly x\]a\jv\\b. Therefore T4 is an LU-subsemilattice of A isomorphic to M4. II. Now, we shall observe the case a\\c. It is evident that then a\\b and c^b. We can suppose Z?||c, otherwise we would obtain the same results as for the case I. (a) First let us suppose aV6<aV6Vc, aVc<aV2>Vc, b\tc-^aMb\Jc. (a) Let L(a,b)=L(a,c)=L(b,c)=0. Then L(U(L(a, c), L(b, c)))= 0, but 244 J. Rachunek L(a\Jb,c)?z0. Let x£L(a\/b,c). Then JR1={X, aVb, aVc, bVc, aVfeVc} is an ZCAsubsemilattice of A isomorphic to N^. (P) If e.g. L(a,b)^0, d£L(a,b), then R2= {d, a\Jb, aVc, bVc, aVbVc} is an L(7-subsemilattice of A isomorphic to Nt. (b) Let a\!b=aVb\lc, aVc^aVb, bVc^aVb. (a) Let L(a, b)—L(a, c)=L(b, c)= 0 . If L(aVc,b)=0, then Rs={a,b, aVc, aV£>} is an LC/-subsemilattice of A isomorphic to M2. If L(aVc,b)?£0, d£L(aVc,b), then Ri= {d, b, aVc, byc, aVb} is an LU-subsemilattice of A isomorphic to M4 . (P) If L(a,b)?±0, e£L(a,b), then R& = {e,b,aMc,b\lc,a\lb} is an LU-sub-semilattice of A isomorphic to Mt. (y) If e.g. L(a, c ) # 0 , f£L(a, c), then Rs= {/, a, aVc, by c, a\lb] is an LU-subsemilattice of A isomorphic to M^. (c) Let us suppose aSJb=a\]c=a\lb\lc, bVc^aVb. (a) Let L(a,b)=L(a,c)=L(b,c)=0. If L(a,bMc)=0, then R7={a,b, bye, aVb} is an Lt/-subsemilattice of A isomorphic to M2. Let L(a, bVc)?£0, g£L(a,bVc). Then L(b,g)=L(c,g)=0. If by g= =c\/g=b\lc, then Rs={b, g, c, fcVc} is an Lt/-subsemilattice of A isomorphic to Ns. If bVg<bVc, then R9={g,byg,byc,ayb,a} is an Lt/-subsem¡lattice of A isomorphic to M4 . 0?) Let L(a, b)^0, h£L(a,b). Then R10= {h, b, byc, ayb, a} is an LU-subsemilattice of A isomorphic to M4 . (Similarly for L(a,c)^0.) (y) Let L(a,b)=L(a,c)=0, L(b,c)^0. If L(a,bVc)=0, then R7 is an Z.i/-sub.semilattice of A. Suppose L(a, b\/c)^ 0 , g£L(b, c), h£L(a, b\Jc). We have hVg^b, hVgl£c, hVg^bVc. Let b^hNg. If hVg^bVc, then J ? u = = {h, hVg, b\/c, aVb, a} is an LU subsemilattice of A isomorphic to M 4 . If hSlg= =bVc, then R12= {g, h, b, bVc} is an ¿[/-subsemilattice of A isomorphic to Af2. (For c</iVf, we can prove similarly.) Let b\\h\lg, c\hMg. If b\/hVg=bVc and cV/iVg=bVc, then R13= = {h, b, hWg, c, b\Jc) is an Li7-subsemilattice of A isomorphic to Nt. If &V/iVg< <fcVc or cV/iVg<&Vc, respectively, then Rlt= {h, c, b, bVhVg, byc} or Ru— = {h, b, c, cVhVg, bVc}, respectively, is an L£/-subsemilattice of ,4 isomorphic to M4 . (d) The case aVc=6Vc=aVfcVc, aV6<aVc can be proved analogously as the case (c). (e) Let us suppose aVfr=aVc=ftVc=aVfeVc. (a) If L(a,b)=L(a,c)—L(b,c)=0, then R1IS= {a, b, c, aVbVc} is an LU-subsemilattice of A isomorphic to N3. (/3) Let e.g. L(a, b)?±0, d£L(a,b). If £?Vc<aVbVc, then R17= {d, a, b, dVc, ay bye} is an LtZ-subsemilattice of A isomorphic to Nt. «-distributive semilattices 245 Let ¿Vc=aVftVc and let L(b,c)=0 or L(a,c)=0, respectively. Then R18={d, b, c, aVbVc) or R'1B = {d, a, c, aMbVc}, respectively, is an L£/-subsemi-lattice of A isomorphic to M2. Finally, let us observe the case L(a, b)^0, L(a, L(b, c ) ? s 0 . Let d£L(a,b),e£L(a,c), fdL(b,c). If e.g. L(e,f)^0, g£L(e,f), then R19={g,a, b,c,aVb\/c} is an LCZ-subsemilattice of A isomorphic to Nt. Hence, let L(d,e)— L(d,f)=L(e,f)=0. Since L(aVb, c)—L{c), it exists (by the assumption) an element U(L(a, c), L(b, c)) such that c^x. For x we have x^e, x = / , thus it must be c>eVf. If now aV/>c, then R20 = {e, a, eVf, c, aVf} is an LU-sub-semilattice of A isomorphic to M4 . Let aV/||c. If aV/>a , then R21={e, a, aV/, c, a\/b\/c} is an ¿CZ-subsemi-lattice of A isomorphic to M4. If aVf=a, then R22= {/, a, b, c, a\Jb\f c} is an L(7-subsemilattice of A isomorphic to Nt. All remaining possibilities of the connections among a, b, c would lead to some variants of the preceding cases only. Remark 4. In [2] it is proved for any ordered set A that if A is non-distributive, then it contains an L£/-subset isomorphic to some of ordered sets M1, M2, M3, Mt, M5, M6, JVls N2, N3, Ni} N5. (See Fig. 2 and 3.) But for the case of semilattices, the constructions of respective L[/-subsets from [2] do not lead to subsemilattices. b< a b c e a o N1 b c o o Fig. 3 2 246 J. Rachiinek: o-distributive semilattices D e f i n i t i o n 3. A subset M of an ordered set A is called strong if for any a,b£M it holds: (i) LA(U„(fl,b)) = LA{UA(a,b)y, (ii) UA(LM(a, b)) = UA(LA(a, b)). In [2] it is shown that if M is a strong subset of A such that UA(a, b)?±{ 1} and La(a, b) JT {0} (where 1 or 0 denotes the greatest or the least element of A, respectively — if they exist), then M is an ¿(/-subset of A. Furthermore, any strong subset of an ordered set A which is a semilattice with respect to the induced order, is a subsemilattice of A. Therefore, the following theorem is similar to the converse of Theorem 2. Theorem 3. If a semilattice A—(A,V) contains an LU-subsemilattice iso-morphic to M2 or to N3, respectively, or if it contains a strong subsemilattice iso-morphic to Af4 or to Na, respectively, then A is non-o-distributive (and so non-dis-tributive, too). Proof . The assertion follows from [2, Theorems 4 and 7]. It is clear that the non-distributivity of A for the cases of the strong subsemilattices M2 and M3 also directly follows from the fact that A is not (in those cases) lower directed. References []] GRATZER, General lattice theory (Russian ed.), Mir (Moscow, 1982). [2] I. CHAJDA and J. RACHUNEK, Forbidden configurations for distributivite and modular ordered sets, Order, 5 (1989), 407—423. [3] J. LARMEROVA and J. RACH£INEK, Translations of modular and distributive ordered sets, Acta Univ. Palack. Olom., Fac. Rer. Nat., Math., 91 (1988), 13—23. DEPARTMENT OF ALGEBRA AND GEOMETRY PALACKY UNIVERSJTY 771 46 OLOMOUC, CZECHOSLOVAKIA 

A characterization of o-distributive semilattices

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A characterization of o-distributive semilattices

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