On join irreducible -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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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 -trivial semigroups () are exhibited with the property that, while each is not join irreducible, the monoid is join irreducible. The monoids are the first examples of join irreducible -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 -trivial semigroup of order up to six are completely described. It turns out that besides known examples and those generated by and its dual monoid, there are no further examples.

Cite this article

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

DOI 10.4171/RSMUP/90