JournalsrsmupVol. 147pp. 43–78

On join irreducible JJ-trivial semigroups

  • Edmond W.  H. Lee

    Nova Southeastern University, Fort Lauderdale, FL, USA
  • John Rhodes

    University of California, Berkeley, USA
  • Benjamin Steinberg

    City College of New York, USA
On join irreducible $J$-trivial semigroups cover
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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 J\mathscr{J}-trivial semigroups Cn\mathcal{C}_n (n2n \geq 2) are exhibited with the property that, while each Cn\mathcal{C}_n is not join irreducible, the monoid CnI\mathcal{C}_n^I is join irreducible. The monoids CnI\mathcal{C}_n^I are the first examples of join irreducible J\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 J\mathscr{J}-trivial semigroup of order up to six are completely described. It turns out that besides known examples and those generated by C2I\mathcal{C}_2^I and its dual monoid, there are no further examples.

Cite this article

Edmond W.  H. Lee, John Rhodes, Benjamin Steinberg, On join irreducible JJ-trivial semigroups. Rend. Sem. Mat. Univ. Padova 147 (2022), pp. 43–78

DOI 10.4171/RSMUP/90