Interpretability Types for Regular Varieties of Algebras

Abstract

It is proved that for every regular variety V of algebras, an interpretability type [V] in the lattice \({\mathbb{L}^{\operatorname{int} }} \) is primary w.r.t. intersection, and so has at most one covering. Moreover, the sole covering, if any, for [V] is necessarily infinite. For a locally finite regular variety V, [V] has no covering. Cyclic varieties of algebras turn out to be particularly interesting among the regular. Each of these is a variety of n-groupoids (A; f) defined by an identity \(f(x_1 ,...,x_n ) = f(x_{\lambda (1)} ,...,x_{\lambda (n)} )\), where λ is an n-cycle of degree n ≥ 2. Interpretability types of the cyclic varieties form, in \({\mathbb{L}^{\operatorname{int} }} \), a subsemilattice isomorphic to a semilattice of square-free natural numbers n ≥ 2, under taking m ∨ n=[m,n] (l.c.m.).