Approximate distributive laws and finite equational bases for finite algebras in congruence-distributive varieties (Q818705)

An operational unary polynomial on an algebra \({A}\) is defined to be a polynomial function obtained by freezing all entries except one in a basic operation on \({A}\). Let \(O_A\) be the set of all operational polynomials of \({A}\). For \(\alpha\in\text{ Eqv}(A)\) let \(\mathcal{O}_{\alpha}\) be the equivalence relation on \(A\) generated by \(\alpha\cup\{\langle p(\alpha),q(\alpha)\rangle:p\in O_A,\langle a,b\rangle\in\alpha\}\). Let \(M\) be a positive integer. An algebra \({A}\) is said to have Mal'tsev depth at most \(M\) if \({\mathcal O}^{M+1}={\mathcal O}^M\) on \(\text{Eqv}(A)\). The authors consider certain conditions dealing with equivalence relations and the operator \(\mathcal{O}\) which they denote as approximate distributive laws. These conditions are applied to give constructive proofs of the following results: (1) Let \(\mathcal{V}\) be a congruence-distributive locally finite variety. If the Mal'tsev depths of finitely subdirectly irreducible members of \(\mathcal{V}\) are bounded by \(N\), then all members of \(\mathcal{V}\) have Mal'tsev depth bounded by \(N+D\), where \(D\) is the maximum depth of the designated Jónsson terms for \(\mathcal{V}\). (2) A congruence-distributive variety of finite type that is generated by a finite algebra is finitely based.