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Minimal monoids generating varieties with complex subvariety lattices

Part of: Varieties Semigroups

Published online by Cambridge University Press:  22 March 2024

Sergey V. Gusev Open the ORCID record for Sergey V. Gusev [Opens in a new window]
Sergey V. Gusev*
Affiliation:
Department of Mathematics, Mechanics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg, Russia (sergey.gusb@gmail.com)
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Abstract

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. We show that the 6-element Brandt monoid generates a finitely universal variety of monoids and, by the previous results, it is the smallest generator for a monoid variety with this property. It is also deduced that the join of two Cross varieties of monoids can be finitely universal. In particular, we exhibit a finitely universal variety of monoids with uncountably many subvarieties which is the join of two Cross varieties of monoids whose lattices of subvarieties are the 6-element and the 7-element chains, respectively.

Keywords

monoidvarietylattice of varietiesfinitely universal varietyBrandt monoid

MSC classification

Primary: 20M07: Varieties and pseudovarieties of semigroups
Secondary: 08B15: Lattices of varieties
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 26
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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