On join irreducible J-trivial semigroups
Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 43-78.
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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 𝒥-trivial semigroups 𝒞 n (n≥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.

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DOI : 10.4171/rsmup/90
Classification : 20, 08 
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     author = {Lee, Edmond W. \hspace{1pt}H. and Rhodes, John and Steinberg, Benjamin},
     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://archive.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://archive.numdam.org/articles/10.4171/rsmup/90/

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