Conjugacy in inverse semigroups

João Araújo; Michael Kinyon; Janusz Konieczny

doi:10.1016/j.jalgebra.2019.05.022

Conjugacy in inverse semigroups

João Araújo, Michael Kinyon, Janusz Konieczny

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Abstract

In a group G, elements a and b are conjugate if there exists g∈G such that g⁻¹ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼_ib, if there exists g∈S¹ such that g⁻¹ag=b and gbg⁻¹=a. The purpose of this paper is to study the conjugacy ∼_i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.

Original language	English
Pages (from-to)	142-173
Number of pages	32
Journal	Journal of Algebra
Volume	533
DOIs	https://doi.org/10.1016/j.jalgebra.2019.05.022
Publication status	Published - 1 Sep 2019

Keywords

Bicyclic monoid
Clifford semigroups
Conjugacy
Factorizable inverse monoids
Free inverse semigroups
Inverse semigroups
McAllister P-semigroups
Stable inverse semigroups
Symmetric inverse semigroups

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1-s2.0-S0021869319302765-main-1Final published version, 522 KBLicence: CC BY-NC-ND

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title = "Conjugacy in inverse semigroups",

abstract = "In a group G, elements a and b are conjugate if there exists g∈G such that g−1ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼ib, if there exists g∈S1 such that g−1ag=b and gbg−1=a. The purpose of this paper is to study the conjugacy ∼i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.",

keywords = "Bicyclic monoid, Clifford semigroups, Conjugacy, Factorizable inverse monoids, Free inverse semigroups, Inverse semigroups, McAllister P-semigroups, Stable inverse semigroups, Symmetric inverse semigroups",

author = "Jo{\~a}o Ara{\'u}jo and Michael Kinyon and Janusz Konieczny",

note = "info:eu-repo/grantAgreement/FCT/5876/147271/PT# The first and second authors were partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matematica e Aplicacoes), the project PTDC/MHC-FIL/2583/2014, the FCT project PTDC/MAT-PUR/31174/2017. The second author was also partially supported by a Simons Foundation Collaboration Grant 359872.",

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N2 - In a group G, elements a and b are conjugate if there exists g∈G such that g−1ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼ib, if there exists g∈S1 such that g−1ag=b and gbg−1=a. The purpose of this paper is to study the conjugacy ∼i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.

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