On join irreducible $J$-trivial semigroups

Abstract

A pseudovariety of semigroups is join irreducible if, whenever it is contained in the complete join of some pseudovarieties, then it is contained in one of the pseudovarieties. A finite semigroup is join irreducible if it generates a join irreducible pseudovariety. New finite $\mathcal{J}$-trivial semigroups ${\mathcal{C}}_n$ ($n\ge 2$) are exhibited with the property that, while each ${\mathcal{C}}_n$ is not join irreducible, the monoid ${\mathcal{C}}_n^I$ is join irreducible. The monoids ${\mathcal{C}}_n^I$ are the first examples of join irreducible $\mathcal{J}$-trivial semigroups that generate pseudovarieties that are not self-dual. Several sufficient conditions are also established under which a finite semigroup is not join irreducible. Based on these results, join irreducible pseudovarieties generated by a $\mathcal{J}$-trivial semigroup of order up to six are completely described. It turns out that besides known examples and those generated by ${\mathcal{C}}_2^I$ and its dual monoid, there are no further examples.