Some varieties containing relation algebras

Abstract

Three varieties of algebras are introduced which extend the variety R A RA of relation algebras. They are obtained from R A RA by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or N A NA , W A WA , and S A SA , respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that W A WA is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that W A WA is the closure of the variety of representable R A RA ’s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.