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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
Received: 08.04.2002

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}


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