# Join irreducible semigroups

## TL;DR

This paper systematically studies finite semigroups that generate join irreducible pseudovarieties, characterizes them, introduces operators preserving this property, and describes all such pseudovarieties generated by small semigroups, expanding existing theory.

## Contribution

It introduces a new operator preserving join irreducibility, characterizes finite semigroups generating join irreducible pseudovarieties, and classifies all such pseudovarieties from semigroups of order up to five.

## Key findings

- Characterization of semigroups generating join irreducible pseudovarieties.
- Introduction of an operator that preserves join irreducibility.
- Complete classification of join irreducible pseudovarieties from small semigroups.

## 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 $A \times B$ of finite semigroups, then $S$ divides either $A^n$ or $B^n$ for some $n \geq 1$. We present a new operator ${ \mathbf{V} \mapsto \mathbf{V}^\mathsf{bar} }$ 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 The $q$-theory of Finite Semigroups, Springer Monographs in Mathematics (Springer, Berlin, 2009).