On join irreducible $J$-trivial semigroups

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 J -trivial semigroups Cn (n ≥ 2) are exhibited with the property that while each Cn is not join irreducible, the monoid C I n is join irreducible. The monoids Cn are the first examples of join irreducible 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 J -trivial semigroup of order up to six are completely described. It turns out that besides known examples and those generated by C2 and its dual monoid, there are no further examples. Mathematics Subject Classification (2010). 20M07, 08B15.