Frames and rings in congruence modular varieties

Abstract

Frames in modular lattices generate rings by von Neumann's extension of the classical representation theorem for projective geometries. In this paper it is shown that frames of congruence relations of algebras in congruence modular varieties generate rings which are also “endomorphism rings” in some sense. To achieve this goal, homomorphisms between Abelian congruences and certain congruences on free algebras are investigated. As an application, we relate the congruence identities of a modular variety to its ring.