Mathematics Faculty Articles

Join irreducible semigroups

Document Type

Article

Publication Date

7-2019

Publication Title

International Journal of Algebra and Computation

Keywords

Semigroup, Pseudovariety, Join irreducible

ISSN

0218-1967

Volume

29

Issue/No.

7

First Page

1249

Last Page

1310

Abstract

We begin a systematic study of finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups S that generate join irreducible pseudovarieties are characterized as follows: whenever S divides a direct product A×B of finite semigroups, then S divides either An or Bn for some n≥1. We present a new operator V↦Vbar that preserves the property of join irreducibility, as does the dual operator, and show that iteration of these operators on any nontrivial join irreducible pseudovariety leads to an infinite hierarchy of join irreducible pseudovarieties. We also describe all join irreducible pseudovarieties generated by a semigroup of order up to five. It turns out that there are 30 such pseudovarieties, and there is a relatively easy way to remember them. In addition, we survey most results known about join irreducible pseudovarieties to date and generalize a number of results in Sec. 7.3 of [Theq-theory of Finite Semigroups, Springer Monographs in Mathematics (Springer, Berlin, 2009)].

Comments

Dedicated to the 80th birthday of Norman Reilly on 30 Jan 2020 and the 65th birthday of Mikhail Volkov on 27 May 2020

Additional Comments

AMSC: 20M07

DOI

10.1142/S0218196719500498

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Peer Reviewed

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