Join irreducible pseudovarieties of dimension six

Abstract

A nontrivial pseudovariety of semigroups is join irreducible if its inclusion in the join of a collection of pseudovarieties implies its inclusion in at least one member of that collection. The dimension of a finitely generated pseudovariety is defined as the order of its minimal generating semigroup. There are precisely $30$ join irreducible pseudovarieties of dimension $5$ or less. While those of dimension $6$ have not been fully identified, $16$ such examples are known so far.
In this article, several pseudovarieties are examined to determine their join irreducible subpseudovarieties, achieving complete identification in some cases. By applying these results alongside existing sufficient conditions, join irreducible pseudovarieties of dimension $6$ are completely identified. Notably, no additional examples exist beyond the $16$ previously known. This is a surprising result, considering that there are $28,634$ semigroups of order $6$ up to isomorphism.