Representation of modular lattices and of relation algebras

Abstract

Introduction. This paper gives an axiomatic characterization of the class of all those (modular) lattices which are isomorphic to lattices of commuting equivalence relations. As might be expected, this problem turns out to be closely related to the representation problem for relation algebras, and we are able to borrow some basic ideas from the work of R. Lyndon [7; 8]. It turns out to be convenient to consider first the class of all those algebras SI = (A, ;, •,, 1') which can be represented isomorphically by means of binary relations over some set U in such a way that the operations ;, •, correspond to relation-theoretic multiplication, set-theoretic multiplication and relationtheoretic conversion, respectively, and 1' corresponds to the identity relation over U. This class of algebras is characterized in Theorem 1. In Theorem 2 this result is applied to lattices. The key observation here is that a lattice St = (A, +, •) with a zero element 0 is isomorphic to a lattice of commuting equivalence relations if and only if the algebra 31 = (A, +, ■,, 0), where 2c~ = x for every xEA, can be represented isomorphically by means of binary relations in the manner discussed above with -fand 0 taking the place of ; and 1'. We can give a more direct, although more involved, proof of Theorem 2. The added complications are due mostly to the fact that we are then unable to take advantage of the metamathematical results of Henkin [2] and Tarski [9]. The trouble is that we see no direct way of proving that the class of lattices under consideration coincides with the class of all subalgebras of algebras from an arithmetic class in the wider sense. The method employed here also has the advantage that it brings out the close connection with the representation problem for relation algebras, and in this context Theorem 1 is of independent interest. Theorems 3 and 4 concern the existence of what we call weak representations for relation algebras, and the paper concludes with a brief discussion of some open questions related to our work. 1. Preliminaries. We adopt the notation and terminology of Tarski [9], in particular, we refer the reader to that paper for the definitions of an arithmetic class and an arithmetic class in the wider sense. The relationtheoretic product of two binary relations and 5 will be denoted by R S, their set-theoretic sum and product by RKJS and RC\S, and the relationtheoretic converse of by R~x. By an algebra of relations we mean a system (a, |, C\, ~ I) where ft is a set of binary relations which is closed under the