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Transactions of the American Mathematical Society

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Monoid varieties with extreme properties

Authors: Marcel Jackson and Edmond W. H. Lee
Journal: Trans. Amer. Math. Soc. 370 (2018), 4785-4812
MSC (2010): Primary 20M07
Published electronically: January 18, 2018
MathSciNet review: 3812096
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Abstract: Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and, surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids with simple descriptions that generate monoid varieties with continuum many subvarieties; these include inherently nonfinitely based finite monoids and all monoids for which $ xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition.

Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross.

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Additional Information

Marcel Jackson
Affiliation: Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

Edmond W. H. Lee
Affiliation: Department of Mathematics, Nova Southeastern University, Fort Lauderale, Florida 33314

Received by editor(s): July 2, 2015
Received by editor(s) in revised form: September 20, 2016, and September 26, 2016
Published electronically: January 18, 2018
Additional Notes: The first author was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666
Dedicated: Dedicated to the 81st birthday of John L. Rhodes
Article copyright: © Copyright 2018 American Mathematical Society