Interpretability Types for Regular Varieties of Algebras

D. M. Smirnov

Algebra i logika, Tome 43 (2004) no. 2, pp. 229-234

Résumé

It is proved that for every regular variety $V$ of algebras, an interpretability type $[V]$ in the lattice ${\mathbb L}^{\rm int}$ 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 $f(x_1,\ldots, x_n)=f(x_{\lambda(1)},\ldots, x_{\lambda(n)})$, where $\lambda$ is an $n$-cycle of degree $n\geqslant 2$. Interpretability types of the cyclic varieties form, in ${\mathbb L}^{\rm int}$, a subsemilattice isomorphic to a semilattice of square-free natural numbers $n\geqslant 2$, under taking $m\vee n=[m,n]$ (l.c.m.).