Join irreducible 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 [Formula: see text] that generate join irreducible pseudovarieties are characterized as follows: whenever [Formula: see text] divides a direct product [Formula: see text] of finite semigroups, then [Formula: see text] divides either [Formula: see text] or [Formula: see text] for some [Formula: see text]. We present a new operator [Formula: see text] 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 [Formula: see text] 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[Formula: see text]-theory of Finite Semigroups, Springer Monographs in Mathematics (Springer, Berlin, 2009)].