Mathematics Faculty Articles

Title

Monoid Varieties with Extreme Properties

Document Type

Article

Publication Date

1-2018

Publication Title

Transactions of the American Mathematical Society

ISSN

1088-6850

Volume

370

Issue/No.

2018

First Page

4785

Last Page

4812

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.

Comments

The first author was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666

Dedicated to the 81st birthday of John L. Rhodes

Additional Comments

© Copyright 2018 American Mathematical Society

DOI

10.1090/tran/7091

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Peer Reviewed

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