Compatible relations of modular and orthomodular lattices

Abstract

Let L be a modular lattice of finite length. L is a projective geometry if and only if L has only trivial tolerances. A binary relation 9 is called a tolerance of an algebra W = (A, Q) if 9 is reflexive, symmetric and compatible with the operations of W. The tolerances D = ((a, a)la E A) and A2 are called the trivial tolerances of W. Obviously every congruence relation of 9I is also a tolerance of W. If L is a lattice then we consider R = ((a, b)la, b E A, a (a, Abl)V(a2A b2) and b, Vb2 > (a, Abl)V(a2 A b2) we have ((a1 V a2) A (b, V b2), b1 V b2) E p. Similarly we prove the three other conditions and have (a1 V a2, b, V b2) E (. In the same way we can show that f is compatible with the operation A. The function s is also order-preserving. We have t o s(p) = t(t) = f n R. If (c, d) E t n R then we have (c, d) = (c A d, d) E p. If (a, b) E p then (a, b) E= R and (a A b, b) E p, (a A b, a) E p, (a, a V b) E p and (b, a V b) E p and therefore (a, b) Et n R. We have t o s = Is and s o t = 1 T is proved similarly. THEOREM 2. Let L be an orthomodular lattice. A binary relation 9 of L is a congruence relation if and only if 9 is reflexive, symmetric and compatible with join and meet. Received by the editors January 21, 1980. 1980 Mathematics Subject Classification. Primary 06C05; Secondary 06C15. ? 1981 American Mathematical Society 0002-9939/81/0000-0126/$01 .75 462 This content downloaded from 207.46.13.189 on Thu, 16 Jun 2016 06:25:00 UTC All use subject to http://about.jstor.org/terms MODULAR AND ORTHOMODULAR LATTICES 463 PROOF. As L is relatively complemented 9 is a lattice congruence of L [4], [7]. It remains to show that from a 9 b we have a' 9 b'. We assume a 0 and a+ = 0 else. In a modular lattice of finite length we have (a V b)+ = a + V b + [3, p. 269, Lemma 6.1(e)]. We consider the following binary relation p = ((a, b)Ia 3. L is isomorphic to the lattice of all subspaces of a vector space over some division ring if and only if L has only trivial tolerances. These results cannot be extended to lattices of infinite length. The restriction is necessary since the relation a p b iff a < b and codimb(a) < oo will generate a proper nontrivial tolerance relation on the subspace lattice of an infinite-dimensional projective space. R-EFERENCES 1. G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. 2. I. Chaida, J. Niederle and B. Zelinka, On existence conditions for comWatible tolerances, Czechoslovak Math. J. 26 (1976), 304-311. 3. G. D. Findlay, Reflexive homomorphic relations, Canad. Math. Bull. 3 (1960), 131-132. 4. J. Hashimoto, Congnrence relations and congruence classes in lattices, Osaka J. Math. 15 (1963), 71-86. 5. C. Hermann, S-verklebte Summen von Verbanden, Math. Z. 130 (1973), 255-274. This content downloaded from 207.46.13.189 on Thu, 16 Jun 2016 06:25:00 UTC All use subject to http://about.jstor.org/terms 464 DIETMAR SCHWEIGERT 6. G. Gratzer, General lattice theory, Stuttgart, 1978. 7. G. Gritzer and E. T. Schmidt, On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13 (1962), 178-185. 8. M. Kamara and D. Schweigert, Eine Charakterisierung polynomvollstuidger Polaritatsverbinde, Arch. Math. (Basel) 30 (1978), 661-664. 9. H. Werner, A Mal'cev condtion for admissible relations, Algebra Universalis 3 (1973), 263. 10. R. Wille, Eine Charakterisierung endlicher, ordnungspoIynomvollst?ndiger VerbEinde, Arch. Math. (Basel) 28 (1977), 557-560. FB MATHMATIK, UNvRSrIAT KsSERsLAuTERN, 675 KmsRsLAurERN, WESm GERMANY This content downloaded from 207.46.13.189 on Thu, 16 Jun 2016 06:25:00 UTC All use subject to http://about.jstor.org/terms