On join irreducible $J$-trivial semigroups
Edmond W. H. Lee, John Rhodes, Benjamin Steinberg · Rendiconti del Seminario Matematico della Università di Padova · 2022
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 \mathscr{J} -trivial semigroups \mathcal{C}_n ( n \geq 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 \mathscr{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 \mathscr{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.