On join irreducible J-trivial semigroups

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

On join irreducible J-trivial semigroups

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

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

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-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 C₂^I and its dual monoid, there are no further examples.

Original language	American English
Pages (from-to)	43–78
Journal	Rendiconti del Seminario Matematico della Università di Padova
Volume	147
State	Published - Jul 2022

Funding

Funding – John Rhodes was supported by Simons Foundation Collaboration Grants for Mathematicians #313548. Benjamin Steinberg was supported by PSC-CUNY.

Funders	Funder number
PSC-CUNY
Simons Foundation	313548

ASJC Scopus Subject Areas

Analysis
Algebra and Number Theory
Mathematical Physics
Geometry and Topology

Keywords

Semigroup
J-trivial
Pseudovariety
Join irreducible

Disciplines

Mathematics

Cite this

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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-trivial semigroups Cn (n ≥ 2) are exhibited with the property that, while each Cn is not join irreducible, the monoid CnI is join irreducible. The monoids CnI 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 C2I and its dual monoid, there are no further examples.",

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AU - Lee, Edmond W. H.

AU - Rhodes, John

AU - Steinberg, Benjamin

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N2 - 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 CnI is join irreducible. The monoids CnI 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 C2I and its dual monoid, there are no further examples.

AB - 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 CnI is join irreducible. The monoids CnI 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 C2I and its dual monoid, there are no further examples.

KW - Semigroup

KW - J-trivial

KW - Pseudovariety

KW - Join irreducible

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ER -

On join irreducible J-trivial semigroups

Abstract

Funding

ASJC Scopus Subject Areas

Keywords

Disciplines

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