E-disjunctive inverse semigroups

Luna Elliott; Alex Levine; James Mitchell

doi:10.1016/j.jalgebra.2025.08.041

E-disjunctive inverse semigroups

Luna Elliott
, Alex Levine
, James Mitchell

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

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Abstract

In this paper we provide an overview of the class of inverse semigroups S such that every non-trivial congruence on S relates at least one idempotent to a non-idempotent; such inverse semigroups are called E-disjunctive. This overview includes the study of the inverse semigroup theoretic structure of E-disjunctive semigroups; a large number of natural examples; some asymptotic results establishing the rarity of such inverse semigroups; and a general structure theorem for all inverse semigroups where the building blocks are E-disjunctive.

Original language	English
Pages (from-to)	292-344
Number of pages	53
Journal	Journal of Algebra
Volume	687
Early online date	17 Sept 2025
DOIs	https://doi.org/10.1016/j.jalgebra.2025.08.041
Publication status	Published - 1 Feb 2026

Keywords

E-disjunctive
Idempotent-pure
Inverse semigroup

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10.1016/j.jalgebra.2025.08.041Licence: CC BY

Elliott_etal_2026_JAlgebraFinal published version, 1.91 MBLicence: CC BY

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abstract = "In this paper we provide an overview of the class of inverse semigroups S such that every non-trivial congruence on S relates at least one idempotent to a non-idempotent; such inverse semigroups are called E-disjunctive. This overview includes the study of the inverse semigroup theoretic structure of E-disjunctive semigroups; a large number of natural examples; some asymptotic results establishing the rarity of such inverse semigroups; and a general structure theorem for all inverse semigroups where the building blocks are E-disjunctive.",

keywords = "E-disjunctive, Idempotent-pure, Inverse semigroup",

author = "Luna Elliott and Alex Levine and James Mitchell",

note = "Data availability statement: No data was used for the research described in the article. Funding information: The second named author was supported by a London Mathematical Society Early Career Fellowship (ECF-2022-09) and the Heilbronn Institute for Mathematical Research during this work.",

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N2 - In this paper we provide an overview of the class of inverse semigroups S such that every non-trivial congruence on S relates at least one idempotent to a non-idempotent; such inverse semigroups are called E-disjunctive. This overview includes the study of the inverse semigroup theoretic structure of E-disjunctive semigroups; a large number of natural examples; some asymptotic results establishing the rarity of such inverse semigroups; and a general structure theorem for all inverse semigroups where the building blocks are E-disjunctive.

AB - In this paper we provide an overview of the class of inverse semigroups S such that every non-trivial congruence on S relates at least one idempotent to a non-idempotent; such inverse semigroups are called E-disjunctive. This overview includes the study of the inverse semigroup theoretic structure of E-disjunctive semigroups; a large number of natural examples; some asymptotic results establishing the rarity of such inverse semigroups; and a general structure theorem for all inverse semigroups where the building blocks are E-disjunctive.

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VL - 687

SP - 292

EP - 344

JO - Journal of Algebra

JF - Journal of Algebra

ER -

E-disjunctive inverse semigroups

Abstract

Keywords

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Diophantine problems in one-relator inverse semigroups

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E-disjunctive inverse semigroups

Abstract

Keywords

Access to Document

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Diophantine problems in one-relator inverse semigroups

Cite this