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.
| Original language | American English |
|---|---|
| Pages (from-to) | 4785–4812 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 370 |
| Issue number | 7 |
| DOIs | |
| State | Published - Jan 1 2018 |
Bibliographical note
Publisher Copyright:© 2018 American Mathematical Society.
Funding
Received by the editors July 2, 2015, and, in revised form, September 20, 2016, and September 26, 2016. 2010 Mathematics Subject Classification. Primary 20M07. The first author was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666. The first author was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666. The authors thank the reviewer for a number of suggestions and for bringing Question 4.2 to their attention.
| Funders | Funder number |
|---|---|
| Australian Research Council | DP1094578, FT120100666 |
| Centre of Excellence for Electromaterials Science, Australian Research Council |
ASJC Scopus Subject Areas
- General Mathematics
Disciplines
- Mathematics