Tomková, Mária

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Mathematica SlovacaMária TomkováGraph isomorphisms of modular multilatticesMathematica Slovaca, Vol. 30 (1980), No. 1, 95--100Persistent URL: http://dml.cz/dmlcz/131090Terms 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. Slova ca 30,1980, No. 1, 95—100 GRAPH ISOMORPHISMS OF MODULAR MULTILATTICES MARIA TOMKOVA 1. Preliminaries A partially ordered set P is said to be of locally finite length if each bounded chain in P is finite. For the elements a, b eP we write a>b (a covers b) if a >b and if there does not exist any element c eP with a >c >b ; in this case the interval [a, b] is called prime. A partially ordered set P is called upper (lower) directed if for each pair of elements a, b eP there exists an element heP (deP) such that a=h, b=h (d = a,d = b). The upper and lower directed partially ordered set is called directed. A multilattice [1] is a partially ordered set M in which the conditions (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 e M, z=v,z=a,z=b implies z=v. A multilattice M is modular [1] iff for every a, b, c, u, v eM satisfying the conditions u=a=v, u=b=c=v, v eavb, u eaAc we have b =c. A multilattice M is distributive [1] iff for every a, b, c, u, v eM satisfying the conditions u=a, b, c=v, v eavb, v eavc, u eat\b, u eaAc we have b =c. It is evident that each partially ordered set of locally finite length is a multilattice [1]. All partially ordered sets dealt with in this note are assumed to be of locally finite length. By a graph G(S) of a subset S czP there is meant the unoriented graph (without multiple edges and loops) whose vertices are elements of S ; two vertices a, b eS are joined by the edge (a, b) iff a >b or b>a. We say that unoriented graphs G(Si) and G(S2) are isomorphic if there exists a bijection q) of Si onto S2 satisfying: (JC, y) is an edge in G(Si) iff (<p(x), qp(y)) is an edge in G(S2). The following two assertions (Ti), (T2) were proved in [1] (4.7.4 and Theorem 4.5). (Ti) A multilattice M of locally finite length is modular iff it fulfils the following covering condition (of) and the condition (o") dual to (or). 95 (a') If a, b, u, v e M such that [u, a], [u, b] are prime intervals and v eavb, then [a,v], [b, v] are prime intervals. (T2) LetCi, C2 be two maximal chains from atob in a modular multilattice M of locally finite length. Then C\, C2 are of the same length. A set S = {a, b, u, v}czM is called an elementary square if a, b are incomparable elements and v>a,v>b,u<a,u<b. Let Mi and M2 be directed multilattices of locally finite length and let m be a graph isomorphism of G(M2) onto G(M2). Let S = {a, b, u, v} be an elementary square in Mi. We shall say that S breaks by the isomorphism q> if either the elements cp(u), <p(v) are covered by q)(a), cp(b) or the elements <p(u), <p(v) cover cp(a) and q)(b). Graph isomorphisms of lattices and multilattices have been studied in the papers [2], [3], [4], [5]. In [2] and [3] the following theorems have been proved: (A) If Li andL2 are lattices of locally finite length such that (i) Lx is modular and (ii) the unoriented graphs G(LX), G(L2) are isomorphic, then the lattice L2 is modular as well. (B) Let Mi and M2 be directed distributive multilattices of locally finite length. Then the following conditions are equivalent: (i) The unoriented graphs G(Mi), G(M2) are isomorphic. (ii) There exist multilattices A, B such that Mi is isomorphic with A xB and M2 is isomorphic with A xB (B is dual to B). In the present paper we shall investigate some questions on graph isomorphisms of multilattices analogous to those that have been dealt with in the papers [2], [3], [4], [5]. 2. Statement of results Theorem 1. If Mi and M2 are directed multilattices of locally finite length such that (i) the unoriented graphs G(MX), G(M2) are isomorphic, (ii) M2 is modular and (iii) Mi is distributive, then the multilattice M2 is distributive as well. Theorem 2. There exist directed finite multilattices Mi and M2 such that (i) the unoriented graphs G(Mi), G(M2) are isomorphic, (ii) Mi is modular, (iii) M2 is not modular. Theorem 3. Let Mi and M2 be directed modular multilattices of locally finite length. Then the following conditions are equivalent: (i) There exist a graph isomorphism q> of G(Mi) onto G(M2) such that no elementary square S c M i breaks by the isomorphism q> and no elementary square S' CLM2 breaks by the isomorphism qp-1. (iii) There are multilattices A, B such that Mi is isomorphic with A xB and M2 is isomorphic with AxB (B is dual to B). 96 3. Proofs of theorems 3.1. P r o o f of T h e o r e m 1. First we recall the definition of the ternary betwenness relation [6] in the directed multilattices. Let a, b, x eM. We say that x is between a and b and write axb if Ф) [(aлx)v(b лx)]x =x, (aлx)л(b лx)c=aлb. Directed multilattices Mi, M 2 are said to be b-equivalent if there exists a bijection / of Mi onto M 2 such that for each triple a,b,x eM the relation axb is equivalent with f(a)f(x)f(b). From Theorems 4.3 and 2.2 of [7] it follows: (*) If Mi and M2 are directed modular multilattices of locally finite length such that the unoriented graphs G(Mi), G(M2) are isomorphic, then Mu M2 are b-equivalent. In [9] the following assertion is proved: (**) Let Mi and M 2 be directed b -equivalent multilattices. If the multilattice Mi is distributive, then M2 is distributive as well. If we assume that Mi and M 2 are directed multilattices of locally finite length such that Mi is distributive and M 2 is modular and G(Mi), G(M2) are isomorphic, then by the assertion (*) the multilattices Mi, M 2 are b -equivalent. Thus from (**) it follows that the multilattice M 2 is distributive. 3.2. Proof of T h e o r e m 2. The partially ordered sets Mi and M 2 in Fig. 1 and Fig. 2 are of the same length 4 and card Mi = card M 2 = 13. It is obvious that Mi and M 2 are directed multilattices. 97 The multilattice M2 is not modular because there exist elements y3, y5 e M2 such that z 2>y 3 , z 2 >y 5 and y 3Ay 5={yi} , where yx is not covered by y3, y5. The modularity of Mi will be verified as follows: We define the height v(x) of an element x eMi as the maximum of lengths of chains between the least element of Mi and x. Let us denote by M(i) the set of elements x eMi with v(x) = i. Then M(0)= {a} , M( l ) = {JCI, JC2, JC3}, M(2) = {yi, yz, y3, y4, y5}, M(3) = {zu Z2, Z3}, M(4) = {/}. It is routine to verify that Mi satisfies the Jordan—Dedekind chain condition and that, whenever i is a positive integer andjc,y eM(i), thenjc Ay eM(i — 1) iff x vy eM(i-\-1). Hence the conditions (o') and (a") are fulfilled and therefore Mi is modular. The multilattices Mi andM2 are defined on the same set M={o,x1, x2, x3, y 1, y2, y3, y4, ys, Z\, z2, z3, / } . Let cp be the identical mapping on M, then q) is a graph isomorphism of G(Mi) onto G(M2). 3.3. For proving Theorem 3 we need some results of the papers [8], [3]. (K) [8] Let A be a quasiordered set. There exists a one-to-one correspondence between the nontrivial direct decompositions of the quasiordered set A into two factors and pairs (Ri9 R2) of nontrivial congruence relations Rl9 R2 on A having the properties: (i) R1R2 = R2R1 (ii) R!uR2 = I, RtnR2 = 0 (I, 0 are the greatest and the least elements of the lattice of all equivalence relations on the set A). (iii) If a, b, c eA, a^c, aR\b, bR2c, then a^b^c. (iv) Let a, b, c, d eA, aRib, cRid, aR2c, bR2d, then from a ^b it follows that c^d and from a^c it follows that b^d. To each couple (RX,R2) with the mentioned properties there corresponds the decomposition A ~A/R1xA/R2 and to each element aeA there corresponds the element (aua2), where ax is the equivalence class under Ri ( i= 1, 2) containing a. In the paper [3] the following two lemmas were proved under the assumption that Mi and M2 are directed distributive multilattices and cp is a graph isomorphism of G(Mi) onto G(M2). Lemma 1. For x, y eMi let u ex Ay, v exvy such that [u, x], [u, y] are prime intervals and let q>(x)«p(u)«p(y). Then q>(x)eq)(u)A(p(v), y(y)ecp(u)v<p(v). Lemma 2. For x, y eMxlet u ex Ay, v exvy, such that [x, v], [y, v] are prime intervals and let cp(x)<q>(v), cp(y)<qp(v). Then q)(w)ecp(jt)Acp(y). From the method of the proof of these lemmas in [3] it follows that they remain valid also when we replace the assumption of the distributivity of the multilattices Mi, M2 by the conditions: (a) Mi, M2 are modular multilattices, (b) no elementary square of Mi breaks by the isomorphism cp and no elementary square of M2 breaks by the isomorphism <p~\ For proving the implication (i)-->(ii) in Theorem (b) in [3] only the modularity of 98 the multilattices Mi, M 2 and the assertion of Lemmas 1 and 2 have been applied. From this it follows that the following assertion is valid. (T3) Let Mi and M2 be directed modular multilattices of locally finite length and let cp be a graph isomorphism of G(Mi) onto G(M2). If no elementary square of Mi breaks by the isomorphism cp and no elementary square of M2 breaks by the isomorphism <p_1, then the condition (ii) from Theorem 3 is valid. Now let us suppose that A, B are modular multilattices f ulfiling the condition (ii) from Theorem 3. Let /i be an isomorphism Mi onto A x B , / 2 a n isomorphism M2 onto A xB and let h be the identical mapping on the underlying set o f A x B (this set is clearly equal to the underlying set of A x B ) . Then the mapping tp = f21hfi is a graph isomorphism of G(Mi) onto G(M2). From the definition cp it follows immediately that fi(x)=f2(cp(x)) for each x eMi. Further there exist relations Ri and R2 on Mi such that A is isomorphic with MJRi, B is isomorphic with Mi/R2 and Ri,R2 fulfils the conditions (i)—(iv) from the assertion (K) (where we take the multilattice Mi instead of the quasiordered set A). Then for each x, y eMi we have: (j) If x<y and cp(x)«p(y) (cp(x)>cp(y)), then x=y(R2) (x=y(Ri)). In fact suppose that x <y and cp(x)<q)(y); then there are elements au a2eA, b\,b2eB with /i (x) = (au bx),fi(y) = (a2, b2). At the same time we have f2(<p(x)) = (ai, bx), /2(q>(y)) = (a2, b2). From x <y it follows that we have either (1) a i < a 2 and bi = b2 or (2) ai = a2 and bi<b2. If (2) were valid, then we would have <p(x)>(p(y), which is a contradiction. Therefore the relation (1) holds. From bi = b2 we obtain x=y(R2). Similarly we can verify that if jc<y and cp(x)><p(y), then *=y(.Ri). Assume that an elementary square (a, b, u, v)czMi would break by the isomorphism cp. Hence either q)(w), op(v) cover the elements q>(a) and <p(b), or q>(u), q>(v) are covered by cp(a) and cp(b). Let us consider the first case (the second case being dual). Then a = u(Ri), b=u(Ri) by (j). Hence a =b(Ri). At the same time a=v(R2), b=v(R2) by (j), and hence a=b(R2). From this it follows according to the property (ii) in the assertion (K) that a=b, which is a contradiction. Similarly we can show that no elementary square of the multilattice M 2 breaks by the isomorphism qT1. Thus we have proved that (ii) implies (i). REFERENCES [1] BENADO, M.: Les ensembles paгtiellement oгdonnées et le théorème de гaffinement de Schгeier, II. Théorie des multistructuгes, Czech. Math. J., 5 (80), 1955, 308—344. [2] JAKUBÍK, J.: Unoriented graphs of modular lattices, Czech. Math. J., 25 (100), 1975,240—246. 99 [3] JAKUBÍK, J.: Grafový izomorfizmus multizväzov, Acta fac. reг. nat. Univ. Comenianae Math., 1, 1956, 255—264. [4] ЯKУБИK, Ӣ.: O гpaфичecкoм изoмopфизмe cгpyктyp, Чexocл. мaт. ж., 4 (79), 1954, 131-142. [5] ЯKУБИK, Ӣ.-KOЛИБИAP, M.: O нeкoтopыx cвoйcтвax пap cтpyктyp, Чexocл. мaт. ж., 4 (79), 1954, 1-27. [6] KLAUČOVÁ, O.: Ь-equivalent multilattices, Math. slov., 26, 1976, 63—72. [7] KOLIBIAR, M.: Übeг metrische Vielverbände, I, Acta Fac. гer. natuг. Univ. Comenianae Math., 4, 1959, 187—203. [8] KOLIBIAR, M.: Übeг direkte Produkte von Relativen, Acta Fac. rer. natuг. Univ. Comenianae Math., 10, 1965, 1—9. [9] TOMКOVÁ, M.: On the b-equivalence of multilattices, Math. slov. 27, 1977, 331—336. Received August 29, 1978 Katedra matematiky Strojníckej fakulty VŠT Švermova 9 040 01 Košice О ГРАФОВОМ ИЗОМОРФИЗМЕ МУЛЬТИСТРУКТУР Мария Томкова Резюме В даной статье доказаны три теоремы о направленных мультиструктурах локально конечной длины. Если графы мультиструктур Ми М2 изоморфны, причём М, дистрибутивна и М2 модулярна, тогда М2 также должна быть дистрибутивна. Однако существуют мультиструктуры М,, М2, графы которых изоморфны, причём Мх модулярна и М2 немодулярна. Третья теорема говорит о условиях, при которых из изоморфизма графов модулярных мультиструктур М ь М2 вытекает существование мультиструктур А, В таких что Мх~АхВ,М2~А хВ (где В является дуальна мультиструктура к В). 100 

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Graph isomorphisms of modular multilattices

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Graph isomorphisms of modular multilattices

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

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