Join irreducible semigroups

We begin a systematic study of those 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 pseudovari-eties are characterized as follows: whenever S divides a direct product A × B of finite semigroups, then S divides either A n or B n for some n ≥ 1. We present a new operator V → V 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 at most 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 pseu-dovarieties to date and generalize a number of results in Chapter 7.