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 Algebra Logika, 2004, Volume 43, Number 2, Pages 229–234 (Mi al67)

Interpretability Types for Regular Varieties of Algebras

D. M. Smirnov

Abstract: It is proved that for every regular variety $V$ of algebras, an interpretability type $[V]$ in the lattice ${\mathbb L}^int$ 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 ${\mathbb L}^int$, a subsemilattice isomorphic to a semilattice of square-free natural numbers $n\geqslant 2$, under taking $m\vee n=[m,n]$ (l.c.m.).

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English version:
Algebra and Logic, 2004, 43:2, 128–131

Bibliographic databases:

UDC: 512.572

Citation: D. M. Smirnov, “Interpretability Types for Regular Varieties of Algebras”, Algebra Logika, 43:2 (2004), 229–234; Algebra and Logic, 43:2 (2004), 128–131

Citation in format AMSBIB
\Bibitem{Smi04} \by D.~M.~Smirnov \paper Interpretability Types for Regular Varieties of Algebras \jour Algebra Logika \yr 2004 \vol 43 \issue 2 \pages 229--234 \mathnet{http://mi.mathnet.ru/al67} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2072573} \zmath{https://zbmath.org/?q=an:1115.08007} \transl \jour Algebra and Logic \yr 2004 \vol 43 \issue 2 \pages 128--131 \crossref{https://doi.org/10.1023/B:ALLO.0000020850.77379.17} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42349090315}