On join irreducible $J$-trivial semigroups

Lee, Edmond W.  H.; Rhodes, John; Steinberg, Benjamin

doi:10.4171/rsmup/90

On join irreducible

J

-trivial semigroups

Lee, Edmond W. H. ; Rhodes, John ; Steinberg, Benjamin

Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 43-78.

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Résumé

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 $𝒞_{n}$ ( $n \geq 2$ ) are exhibited with the property that, while each $𝒞_{n}$ is not join irreducible, the monoid $𝒞_{n}^{I}$ is join irreducible. The monoids $𝒞_{n}^{I}$ 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 $𝒞_{2}^{I}$ and its dual monoid, there are no further examples.

Accepté le : 2020-03-24
Publié le : 2022-05-20

MR Zbl

DOI : 10.4171/rsmup/90

Classification : 20, 08

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     title = {On join irreducible $J$-trivial semigroups},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {43--78},
     volume = {147},
     year = {2022},
     doi = {10.4171/rsmup/90},
     mrnumber = {4450783},
     zbl = {1506.20089},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/90/}
}

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Lee, Edmond W.  H.; Rhodes, John; Steinberg, Benjamin. On join irreducible $J$-trivial semigroups. Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 43-78. doi : 10.4171/rsmup/90. http://www.numdam.org/articles/10.4171/rsmup/90/

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