EQUIVALENCE OF M-SYMMETRY AND SEMIMODULARITY IN LATTICES CHINTHAYAMMA MALLIAH AND S. PARAMESHWARA BHATTA 1. Introduction The notion of M-symmetry in lattices is an implication in the language of type <2, 2> while the concept of semimodularity is an implication involving the covering relations. M-symmetry implies semimodularity in all lattices but not conversely since semimodularity could be valid vacuously (that is, due to lack of any coverings in the lattice). In this regard G. Gratzer [2] posed the following problem. PROBLEM ([2, Problem IV. 16]). Is M-symmetry equivalent to semimodularity for algebraic lattices? A negative solution to this problem is given in Section 2 by constructing a counter-example (Theorem 2.1). In the Main Theorem (Theorem 3.1) of Section 3, the equivalence of M-symmetry and semimodularity in the case of weakly atomic continuous lattices is established. As a consequence certain sufficient conditions to make algebraic semimoduiar lattices M-symmetric are deduced. This paper forms a part of the Doctoral thesis of the second author. The authors wish to thank the referee for suggesting some modifications in the presentation of the paper. DEFINITION 1.1 [2]. A pair (a, b) of elements of a lattice L is called modular, in notation, aMb, if x < b implies that x V (a A b) = (x V a) A b. DEFINITION 1.2 [2]. A lattice L is said to be M-symmetric if aMb implies that bMa for any a,beL. For additional information the books [1], [2] and [3] may be referred to. 2. A counter-example The following lemma gives new descriptions of modular pairs of elements in lattices and it is a basis for our further discussions. LEMMA 2.1. The following conditions are equivalent for any pair (a, b) of elements of a lattice L. (1) aMb, Received 8 March 1985; revised 4 November 1985. 1980 Mathematics Subject Classification 06C10. Second author's research supported by CSIR, New Delhi, India, under the Research Fellowship Scheme. Bull. London Math. Soc. 18 (1986) 338-342