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Sib. Èlektron. Mat. Izv., 2019, Volume 16, Pages 175–186 (Mi semr1048)  

Mathematical logic, algebra and number theory

On modular and cancellable elements of the lattice of semigroup varieties

D. V. Skokov, B. M. Vernikov

Ural Federal University, Institute of Natural Sciences and Mathematics, 51, Lenina str., Ekaterinburg, 620000, Russia

Abstract: We continue a study of modular and cancellable elements in the lattice SEM of all semigroup varieties. In 2007, the second author completely determined all commutative semigroup varieties that are modular elements in SEM. In 2018 the authors jointly with S.V.Gusev proved that, within the class of commutative varieties, the properties to be modular and cancellable elements in SEM are equivalent. The objective of this article is to verify that, within some slightly wider class of semigroup varieties, this equivalence is not the case. To achieve this goal, we completely classify semigroup varieties satisfying a permutational identity of length 3 that are modular elements in SEM. Further, we specify a variety with these properties that is not a cancellable element in SEM.

Keywords: semigroup, variety, lattice of varieties, permutational identity, modular element of a lattice, cancellable element of a lattice.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.6018.2017/8.9
Russian Foundation for Basic Research 18-31-00443
17-01-00551_
Both the authors were supported by the Ministry of Education and Science of the Russian Federation (project 1.6018.2017/8.9) and by Russian Foundation for Basic Research (the first author by grant 18-31-00443; the second author by grant 17-01-00551).


DOI: https://doi.org/10.33048/semi.2019.16.010

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Bibliographic databases:

UDC: 512.532.2
MSC: 20M07
Received March 26, 2018, published February 6, 2019
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Citation: D. V. Skokov, B. M. Vernikov, “On modular and cancellable elements of the lattice of semigroup varieties”, Sib. Èlektron. Mat. Izv., 16 (2019), 175–186

Citation in format AMSBIB
\Bibitem{SkoVer19}
\by D.~V.~Skokov, B.~M.~Vernikov
\paper On modular and cancellable elements of the lattice of semigroup varieties
\jour Sib. \`Elektron. Mat. Izv.
\yr 2019
\vol 16
\pages 175--186
\mathnet{http://mi.mathnet.ru/semr1048}
\crossref{https://doi.org/10.33048/semi.2019.16.010}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000462268100010}


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