Representations of relatively free profinite semigroups, irreducibility, and order primitivity
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- by Jorge Almeida and Ondřej Klíma PDF
- Trans. Amer. Math. Soc. 373 (2020), 1941-1981 Request permission
Abstract:
We establish that, under certain closure assumptions on a pseudovariety of semigroups, the corresponding relatively free profinite semigroups freely generated by a nonsingleton finite set act faithfully on their minimum ideals. As applications, we enlarge the scope of several previous join irreducibility results for pseudovarieties of semigroups, which turn out to be even join irreducible in the lattice of pseudovarieties of ordered semigroups, so that, in particular, they are not generated by proper subpseudovarieties of ordered semigroups. We also prove the stronger form of join irreducibility for the Krohn-Rhodes complexity pseudovarieties, thereby solving a problem proposed by Rhodes and Steinberg.References
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Additional Information
- Jorge Almeida
- Affiliation: CMUP, Dep. Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
- MR Author ID: 208246
- Email: jalmeida@fc.up.pt
- Ondřej Klíma
- Affiliation: Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
- Email: klima@math.muni.cz
- Received by editor(s): February 13, 2017
- Received by editor(s) in revised form: July 12, 2019
- Published electronically: December 4, 2019
- Additional Notes: The first author was partially supported by CMUP (UID/ MAT/00144/2019), which is funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020.
The second author was partially supported by the Grant 15-02862S of the Grant Agency of the Czech Republic. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 1941-1981
- MSC (2010): Primary 20M05, 20M07, 20M30; Secondary 20M35
- DOI: https://doi.org/10.1090/tran/7951
- MathSciNet review: 4068286