INTERPRETABILITY TYPES FOR REGULAR VARIETIES OF ALGEBRAS

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Abstract

It is proved that for every regular variety V of algebras, an interpretability type [V] in the lattice ℒint 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(x1,…, xn) = f(xλ(1),…, xλ(n)), where λ is an n-cycle of degree n ≥ 2. Interpretability types of the cyclic varieties form, in ℒint, a subsemilattice isomorphic to a semilattice of square-free natural numbers n ≥ 2, under taking m ∨ n = [m, n] (l.c.m.).

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