Some Varieties Containing Relation Algebras
Roger D. Maddux · Transactions of the American Mathematical Society · 1982
Three varieties of algebras are introduced which extend the variety $RA$ of relation algebras. They are obtained from $RA$ by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or $NA$, $WA$, and $SA$, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that $WA$ is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that $WA$ is the closure of the variety of representable $RA$’s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.