Monoid varieties with extreme properties
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- by Marcel Jackson and Edmond W. H. Lee PDF
- Trans. Amer. Math. Soc. 370 (2018), 4785-4812 Request permission
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
- Email: m.g.jackson@latrobe.edu.au
- Edmond W. H. Lee
- Affiliation: Department of Mathematics, Nova Southeastern University, Fort Lauderale, Florida 33314
- Email: edmond.lee@nova.edu
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4785-4812
- MSC (2010): Primary 20M07
- DOI: https://doi.org/10.1090/tran/7091
- MathSciNet review: 3812096
Dedicated: Dedicated to the 81st birthday of John L. Rhodes