Varieties of relation algebras

Abstract

formulas 4' and 4, *= 1 are equivalent. All these facts were known to Tarski as early as the 1940's, and they have been used by him and his collaborators. However, much of the work in this area was done at a time when the general theory of varieties was in its early stages, and many of the ideas, techniques and results from that theory were therefore not available. It is the purpose of this paper to re-examine and extend some of the known facts about relation algebras, making use of these more recent developments. Our primary concern will be varieties of relation algebras. In particular, we shall give simple equational bases for several interesting varieties, and prove a number of theorems about the lattice of all varieties of relation algebras. Among other things it will be shown that this lattice has infinitely many dual atoms, the conjugate varieties of the full relation algebras on finite sets, and that these varieties have simple equational bases (Theorems 7.5 and 7.7). In order to make this paper more nearly selfcontained, considerable space has been devoted to a summary of known results about relation algebras. It is hoped