Weakly associative relation algebras with polyadic composition operations (Q1595918)

The class WA of weakly associative relation algebras is a finitely axiomatized variety of Boolean algebras with operators. The equational theory of WA is decidable (a result of \textit{I. Németi} [Proc. Am. Math. Soc. 100, 340-344 (1987; Zbl 0638.03061)] and WA is canonical, that is, WA is closed under the formation of perfect extensions. \textit{R. Maddux} [Trans. Am. Math. Soc. 272, 501-526 (1982; Zbl 0515.03039)] proved that every algebra in WA is relatively representable, that is, isomorphic to a subalgebra of the algebra of all subrelations of a symmetric and reflexive binary relation. This paper introduces a related variety, called \(\text{WA}^\infty\), which is obtained from WA by adding an infinite sequence of operations with increasing rank that generalize binary relative multiplication, together with natural axioms for these operations that generalize the weak associative law of WA. The main results are that the equational theory of \(\text{WA}^\infty\) is decidable, \(\text{WA}^\infty\) is a canonical variety, and every algebra in \(\text{WA}^\infty\) is relatively representable.