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 Algebra Logika, 2004, Volume 43, Number 1, Pages 3–31 (Mi al55)

A Weaker Version of Congruence-Permutability for Semigroup Varieties

B. M. Vernikov

Ural State University

Abstract: Congruences $\alpha$ and $\beta$ are 2.5-permutable if $\alpha\vee\beta=\alpha\beta\cup\beta\alpha$, where $\vee$ is a union in the congruence lattice and $\cup$ is the set-theoretic union. A semigroup variety $\mathcal V$ is $fi$-permutable ($fi$-2.5-permutable) if every two fully invariant congruences are permutable (2.5-permutable) on all $\mathcal V$-free semigroups. Previously, a description has been furnished for $fi$-permutable semigroup varieties. Here, it is proved that a semigroup variety is $fi$-2.5-permutable iff it either consists of completely simple semigroups, or coincides with a variety of all semilattices, or is contained in one of the explicitly specified nil-semigroup varieties. As a consequence we see that (a) for semigroup varieties that are not nil-varieties, the property of being $fi$-2.5-permutable is equivalent to being $fi$-permutable; (b) for a nil-variety $\mathcal V$, if the lattice $L(\mathcal V)$ of its subvarieties is distributive then is $fi$-2.5-permutable; (c) if $\mathcal V$ is combinatorial or is not completely simple then the fact that $\mathcal V$ is $fi$-2.5-permutable implies that $L(\mathcal V)$ belongs to a variety generated by a 5-element modular non-distributive lattice.

Keywords: variety, semilattice, nil-semigroup, congruence-permutability.

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English version:
Algebra and Logic, 2004, 43:1, 1–16

Bibliographic databases:

UDC: 512.532.2

Citation: B. M. Vernikov, “A Weaker Version of Congruence-Permutability for Semigroup Varieties”, Algebra Logika, 43:1 (2004), 3–31; Algebra and Logic, 43:1 (2004), 1–16

Citation in format AMSBIB
\Bibitem{Ver04} \by B.~M.~Vernikov \paper A Weaker Version of Congruence-Permutability for Semigroup Varieties \jour Algebra Logika \yr 2004 \vol 43 \issue 1 \pages 3--31 \mathnet{http://mi.mathnet.ru/al55} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2073443} \zmath{https://zbmath.org/?q=an:1115.20048} \transl \jour Algebra and Logic \yr 2004 \vol 43 \issue 1 \pages 1--16 \crossref{https://doi.org/10.1023/B:ALLO.0000015127.50736.36} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42349103042} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Vernikov BM, “Semigroup varieties with 1.5-permutable fully invariant congruences on their free objects”, Acta Applicandae Mathematicae, 85:1–3 (2005), 313–318
2. L. N. Shevrin, B. M. Vernikov, M. V. Volkov, “Lattices of semigroup varieties”, Russian Math. (Iz. VUZ), 53:3 (2009), 1–28
3. B. M. Vernikov, V. Yu. Shaprynskii, “Tri oslablennykh varianta kongruents-perestanovochnosti dlya mnogoobrazii polugrupp”, Sib. elektron. matem. izv., 11 (2014), 567–604
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