Tomková, Mária

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Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaMária TomkováOn a local property of the unoriented graph of a modular multilatticeMathematica Slovaca, Vol. 30 (1980), No. 3, 289--298Persistent URL: http://dml.cz/dmlcz/136246Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1980Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slovaca 30,1980, No. 3, 289—298 ON A LOCAL PROPERTY OF THE UNORIENTED GRAPH OF A MODULAR MULTILATTICE MARIA TOMKOVA L. R. Alvarez [1] investigated unoriented graphs of modular lattices of finite length. In this note an analogous question for modular multilattices will be studied. A partially ordered set P is said to be of locally finite length if each bounded chain in P is finite. Throughout this paper P will be a partially ordered set of locally finite length. For the elements a,b eP we write a >b or b <a (a covers b, or b is covered by a) if a >b and there does not exist any element ceP with a >c >b. If two elements a, beP are noncomparable, we write a\b. A subset S of a partially ordered set P is called a saturated subsystem of P if for each x, y e S we have x <y in P, whenever x <y in S. By a graph G(S) of a subset SczP there is meant the unoriented graph (without multiple edges and loops) whose vertices are elements of S ; two vertices a, b are joined by the edge (a,b) iff either a covers b or b covers a. In such a case we also say that a and b are neighours. A circuit in the graph G(S) is a sequence {(x0,Xi), (jti,Jt2), ..., (xn-uxn)} of distinct edges such that x0 = xn. A graph isomorphic with the graph in Fig. la is called a graph of type K'. A graph isomorphic with the graph in Fig. lb is said to be a cube. L. R. Alvarez [1] proved the following result: (A) Let Lbea modular lattice of finite length. Let FczL and let G(F) be of type K'. Then there exists an element xeL such that G(FKJ{X}) is a cube. In this paper we shall show that for modular multilattices of locally finite length the analogous assertion fails to be true. We shall investigate the question under what additional conditions for F the assertion remains valid. We recall the basic definition concerning multilattices [2]. A multilattice is a poset M in which condition (i) and its dual (ii) are satisfied: (i) If a, b, h eM and a^-h, b^h, then there exists v eM such that (a) v^h, v§a, v=^b and (b) z eM, z=a, z = b, z = v implies z = v. The symbol (avb)h designates the set of all elements v eM satisfying (i); the symbol (aAb)d has a dual meaning. We denote avb = [J(avb)h, aAb= [J(aAb)d. h&a dXa h^b dSb 289 Recall that the sets avb or a/\b may be empty. If M is directed, then avb + 0±a/\b for each pair a, b eM. A multilattice M is said to be modular iff for every a, b, c, d, h eM satisfying d^a, b, c^h, a^b he(avc)h, de(bAc)d we have a = b. We shall describe all nonisomorphic types of partially ordered sets F, card F = 1 satisfying the following conditions: (a) G(F) is of type K'; (b) there exists a directed modular multilattice MF of locally finite length such that F is a saturated subsystem of MF. We denote by & the set of all nonisomorphic types of partially ordered sets as above. Let F e f . We shall say that F can be extended to a cube when it fulfils the following condition: Whenever M is a directed modular multilattice of locally finite length such that F is a saturated subsystem of M, then there exists an element xeM, such that G(FU{JC}) is a cube. We shall determine the set of all partially ordered sets Fe:J that can be extended to a cube (Theorems 3, 4). Assume that M is a directed modular multilattice of locally finite length and F is a saturated subsystem of M such that the graph G(F) contains a circuit K6 = = {(xo, JCI), (xu x2),..., (x4, x5), (x5, Xo)} and the vertexx6, which is joined by edges with three vertices of the circuit K6 such that there exist no triangles in G(F). Let us denote K = {x0, xu ...., x5}. In what follows the elements of K will be denoted by Xt, where i is an integer taken modulo 6. In the Lemmas 1—4 we suppose that G(K) is a circuit. (We do not use the property of the set F.) 290 Lemma 1. The partially ordered set K has at most three minimal elements. Proof. If we assume that K has more than three minimal elements, then some two of them would be comparable and this is a contradiction. Lemma 2. Let the saturated subsystem K have the least element. Then K is isomorphic to the partially ordered set in Fig. 2. Proof. Assume that JC, is the least element of K. Since JC, is a neighbour of the vertices xi+1, xi+5, we have xi+1 >xt, xi+5 >xt. The element xi+1 is joined by an edge with the vertex xi+2 and we shall show that xi+2 >xi+1 is valid. Let xi+2 <xi+1. Since JC, is the least element of K, we have JC,+2>JCJ contradicting JC,+I>JC,. By the same reasoning we get xi+4>xi+5. Now we will show that JC,+I|JC,+4. Assume that we would have JC,+I<JC,+4. Then there exists an element zeM such that xi+1 <z <xi+4. This yields a contradiction, because the multilattice M is modular and hence all maximal chains from JC, to xi+4 must be of the same length. Similarly we cannot have JC,+1>JC,+4; thus JC,+I|JC,+4. Analogously we werify that the relation xi+2\xi+5 holds. This implies that xi+4\xi+2 and hence for the element xi+3, which is joined by edges with JC,+4, xi+2, we have the following possibility: a) jc,+3 covers xi+4 and xi+2. b) jc,+3 is covered by jt,+4 and xi+2. We shall show that the case b) is impossible. In fact, if JC,+3 <xi+4 and JC,+3 <xi+2, then either JC,+3|JC, (and hence xt would not be the least element of K), or xi+3>xt, which would imply that the maximal chains from JC, to JC,+2 are not of the same length. Thus xi+3>xi+2, jt,+3>Jt,+4 and the proof is complete. Lemma 3. Let the subsystem K have two minimal elements. Then it is isomorphic with some of the partially ordered sets in Figs. 3a, 3b, 3c. 291 XL+Ь *-•* Xi*i ÓXí*2 Xi Xиг ^ X І Ü X І O Fig. Зa Fig. Зb X ó * Proof. We may suppose that one of the pairs {JC,, Jci+2}, {JC,, jci+3} is the pair of minimal elements of K. If jct, jci+2 are minimal, then jci<jci+i, jcf<jci+5, JCI+2<JC1+3, jci+2<jci+i. We shall show that in this case jci+3|jci+5. Since the multilattice M is directed and modular, there exists an element u e M such that u <xi9 u <xi+2. If we assume jci+3<jci+5, then the maximal chains from u to Jci+5 are not of the same length, contradicting the modularity of M. For the same reason the relation jci+5<Jci+3 is impossible. This implies that for the element jci+4 we have to investigate the following cases: 3a) jci+4 covers the elements jci+3, jci+5; 3a') jci+4 is covered by jci+3, Jci+5. The case 3a') cannot occur because the element jci+4 is not minimal. In the case 3a) we have obviously JCi+i|jci+3, JCi+i|jci+5. From the Jordan—Dedekind chain condition it follows JCi+i|jci+4, JC,|xi+3, Jci+2|jci+5. Hence in this case the subsystem K is isomorphic with the partially ordered set in Fig. 3a. If JC,, Jci+3 are minimal elements, then JCI<JC1+I, JC,<JCI+5, x i+3<jc i+2, JCi+3<.ri+4. Since Jci+4, JCi+5 are joined by an edge and so are jci+i, Jci+2, we have the following four possibilities: 292 3b0 x i+4>jc i+5 and jci+1>jci+2 3b2) xi+4>xi+5 and jc i+2>x i+i 3bJ) jci+4<jci+5 and Jci+i<Jc,+2 3b2) jci+4<jci+5 and jci+2<jci+i. In all these cases we have evidently .ri+2|xi+4, xi+1\xi+5. In the case 3bi) Xt\xi+2, Jci+5|jci+3 is valid and we shall show JCi+i|xi+5. The assumption xi+1>xi+5 would contradict jci+4>jci+3 and Jci+5<JCi+2 implies jci+5<jci+1, which contradicts .ri+5|xi+i. Analogously we get jci+i|jci+4. Thus in the case 3bi) the subsystem K is isomorphic with the partially ordered set in Fig. 3b. In the case 3b{) we arrive at the same conclusion. In the case 3b2) we have: JCf|jci+3, Jci+3|jci+5. From the modularity of the multilattice M it follows that jci+i|jci+4, jci+5|jci+2. This implies that the subsystem K is isomorphic with the partially ordered set in Fig. 3c. The case 3b2) yields the same conclusion. The following assertion is obvious. Lemma 4. Let the subsystem K have three minimal elements. Then K is isomorphic with the partially ordered set in Fig. 4. Fig. 4 The lemmas 1—4 may be summarized as follows. Theorem 1. Let Mbea directed modular multilattice of locally finite length and let K = {jc0, xu ..., JC5} be a saturated subsystem of Msuch that G(K) is a circuit. Then K is isomorphic with some of the partially ordered sets in Figures 2, 3a, 3b, 3c, 4. Now we shall suppose that JC6 is an element of the multilattice M such that G(Ku{x6}) is of type K' (i.e., Ku{x6}=Fe&). Lemma 5. The subsystem K cannot be isomorphic with the partially ordered set in Fig. 3b. Proof. Assume that the K would be isomorphic with the partially ordered set in Fig. 3b. Then there exists exactly one minimal element JC, of K that is a neighbour of JC6. Thus JC6 is a neighbour of Jci+2, and JC6 is a neighbour of Jci+4. Since xi+4>xt, we have jci<Jc6<Jci+4. From Jcf|jci+2 it follows JCI+2<JC6. This implies jci+2<Jci+4 con-tradicting JCi+2|jCi+4. 293 Lemma 6. Let the subsystem K be isomorphic with the partially ordered set in Fig. 2. Then the partially ordered sets F and that in Fig. 5 are isomorphic or dually isomorphic. Xi*3 Proof. Assume that JC, is a minimal element of K and that JC6 is joined by edges with jci+i, jci+3, JC/+5. From jc i+3>jc i+i, jc i + 3>jc i + 5 and from the modularity of M it follows jc i +i<Jc6<Jc i + 3, jc i + 5<Jc6<Jc i + 3. We have obviously JC6|JCI+2, JC6|JCI+4. Hence the subsystem F is isomorphic with the partially ordered set in Fig. 5. If we assume that JC6 is joined with the elements JC,, JCI+2, Jci+4 and that JC, is a minimal element of K, then by a similar argument we arrive at the conclusion that the subsystem F is dually isomorphic with the partially ordered set in Fig. 5. Lemma 7. Let the subsystem K be isomorphic with the partially ordered set in Fig. 3a (3c). Then the subsystem Fis isomorphic (dually isomorphic) with the some of the partially ordered sets in Figs. 6a, 6b, 6c. Xi*5 X І O * t « 2 Fig. 6a 294 Xt.,* Xi«5 Xi*5 Fig. 6c Proof. Let xi9 xi+2 be minimal elements of K. First we suppose that the element x6 is joined by edges with the elements xi+l, xi+3, xi+5. Since JCI+1|JC1+3, JCI+3|JC<+5, either x6 covers xi+l, xi+3, xi+5 or x6 is covered by xi+l, xi+3, xi+5. In the first case we get x6\xi+4. Hence the subsystem F is isomorphic with the partially ordered set in Fig. 6a. In the second case we have x6\xi9 x6\xi+2 and hence the subsystem F is isomorphic with the partially ordered set in Fig. 6b. If we assume that x6 is joined by edges with the elements xi9 xi+2, xi+4, then from xi+4>xt, JCI+4>JCI+2 and from the modularity of M it follows xt <x6<xi+4. Moreover JC6|JC1+5, JC6|JCI+1, JC6|JC1+3. Hence the subsystem F is isomorphic with the partially ordered set in Fig. 6c. Since the partially ordered sets in Fig. 3c, 3a are dually isomorphic, the assertion concerning the partially ordered set in Fig. 3c can be proved by the dual way. Lemma 8. Let the subsystem K be isomorphic with the partially ordered set in Fig. 4a. Then the subsystem Fis isomorphic or dually isomorphic with some of the partially ordered sets in Figs. 7a, 7b. X i . j X, Xi-ti Xç Fig. 7b X».,4r 2 9 5 Proof. Let xt, xi+2, xi+4 be minimal elements of K. The element x6 can be joined by edges with either xt, xi+2, xi+4 or with xi+l, xi+3, xi+5. Since JCI|XI+2|JCI+4|JCI and ^i+i|x,+3|xl+5|jcI+i, in the first case the element x6 either covers xi9 xi+2, xi+4 or x6 is covered by xi9 xi+2, xi+4. If x6 is joined with xi+l, xi+3, xi+5, we obtain an analogous result. Thus the proof is complete. The lemmas 5—8 may be summarized as follows. Theorem 2. Let M be a directed modular multilattice of locally finite length. If M contains the saturated subsystem FeZF, then F is isomorphic or dually isomorphic with some of the following partially ordered sets in Figs. 5, 6a, 6b, 6c, 7a, 7b. A graph which we obtain from the graph in Fig. 5 by adding of one vertex x7 and edges (x7,xi+2), (x7,xi+4), (x7,Xi), (x7,x6) will be called a cube with a diagonal. Theorem 3. Let M be a directed modular multilattice of locally finite length. If the saturated subsystem F e2F belonging to M is isomorphic or dually isomorphic with the partially ordered set in Fig. 5, then F can be extended either to a cube or to a cube with a diagonal. Proof. Let us assume that F is isomorphic with the partially ordered set in Fig. 5. Since JCI+2<JCI+3, JCI+4<JCI+3 and JC, <JCI+2, JC, <xi+4, from the modularity of M it follows that there exists an element x7eM such that x7e(xi+2Axi+4)Xi, x7<xi+2, x7<xi+4, Xi <x7. Moreover x7 is not identical with any element of F, because F is a saturated subsystem of M. If x7>x6, then from JC, <x6 and from x( <x7 we obtain a contradiction. Thus either x7 is noncomparable with x6 or x7 < x6. In the first case G(Fu{x7}) is a cube and in the second one G(Fu{x7}) is a cube with a diagonal.. In the case when F is dually isomorphic with the set in Fig. 5 the proof is analogous. A multilattice M is said to be distributive [2] iff for every a, b, b', d, heM S3Ltisfymgd^a,b,b'^h,h = (avb)h = (avb')hd = (aAb)d = (a Ab')d we have b=b'. Corollary 1. Let M be a directed distributive multilattice of locally finite length. If the saturated subsystem Fe2F belonging to Mis isomorphic or dually isomorphic with the partially ordered set in Fig. 5, then F can be extended to a cube. The proof follows directly from the Theorem 3 because from the distributivity of M we obtain x6\x7. Corollary 2. Let Mbea directed modular multilattice of locally finite length and let a saturated subsystem F e 2Fbelonging to M be isomorphic or dually isomorphic with the partially ordered set in Fig. 5. If G(M) does not contain any subgraph isomorphic to a cube with a diagonal, then F can be extended to a cube. Theorem 4. Let Fe9*bea partially ordered set isomorphic or dually isomorphic to some of the partially ordered sets in Figs. 6a, 6b, 6c, 7a, 7b. Then there exists 296 a directed modular multilattice M such that F is a saturated subsystem of M and F can be extended neither to a cube nor to a cube with a diagonal Proof. The multilattices Mu M2 and M3 in Fig. 8a, 8b, 8c are modular (the modularity of these multilattices can be proved analogously as in the case of the У. Fig. 8b % multilattice Mi in [3]). The following saturated subsystems of Mi (i = 1, 2, 3) can be extended neither to a cube nor to a cube with a diagonal: {xu x3, yi, y2, y3, Zi, z2} czMt {xu x2, x3, yu y2, y3, Zi) c M -{xu x2, yu y2, y3, y4, z2} cM2 {yuy2,y4,\zuz2,z3,i }<=M3 {yu y2, y3, y4, zi, z2, z3} c=M3. 297 REFERENCES [1] ALVAREZ, L. R.: Undiгected graphs realizable as graphs of modular lattices, Canadian Journal of Mathematics, XVII, 1965, 923—932. [2] BENADO, M.: Les ensembles partiellement ordonnés et le théorème de raffinement de Schгeieг, II Théorie des multistгuctures, Czech. Math. J., 5 (80), 1955, 308—344. [3] TOMKOVÁ, M.: Gгaph isomoфhisms of modular multilattices, Math. Slovaca (to appear). Received September 23, 1978 Katedra matematiky Strojníckej fakulty VŠT Švermova 9 040 01 Košice ОД ОДНОМ ЛОКАЛЬНОМ СВОЙСТВЕ НЕОРИЕНТИРОВАННОГО ГРАФА МОДУЛЯРНЫХ МУЛЬТИСТРУКТУР Мария Томкова Резюме Л. Р. Алварез доказал что если подфаф О(Р) куба, который получится из куба удалением одной вершины и ребер с ней инцидентных является подфафом неориентированного фафа С (Ь) модулярной структуры ^, которая имеет конечную длину, потом весь куб является подфафом фафа этой структуры. В этой статье доказано, что для модулярных направленных мультиструк­тур аналогическое утверждение не имеет место а изучается вопрос, при которых дополнитель­ных условиях для частично упорядоченного множества Р утверждение правдиво. 298 

On a local property of the unoriented graph of a modular multilattice

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On a local property of the unoriented graph of a modular multilattice

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Institute of Mathematics AS CR, v. v. i.