The lattice of one-sided congruences on an inverse semigroup

Matthew Brookes

doi:10.1007/s10998-022-00497-z

The lattice of one-sided congruences on an inverse semigroup

Matthew Brookes^*

^*Corresponding author for this work

Pure Mathematics

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

5 Downloads (Pure)

Abstract

We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of the trace and inverse kernel are discussed, in particular that the inverse kernel is a full inverse subsemigroup and that both the trace and inverse kernel maps are onto ∩-homomorphisms. It is shown that a left congruence is determined by its trace and inverse kernel, and the lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We demonstrate that every finitely generated left congruence is the join of a finitely generated trace minimal left congruence and a finitely generated idempotent separating left congruence. Characterisations are given of inverse semigroups that are left Noetherian, or are such that Rees left congruences are finitely generated.

Original language	English
Pages (from-to)	1-26
Number of pages	26
Journal	Periodica Mathematica Hungarica
Volume	87
Issue number	1
Early online date	13 Dec 2022
DOIs	https://doi.org/10.1007/s10998-022-00497-z
Publication status	Published - 1 Sept 2023

Keywords

Inverse semigroup
Inverse monoid
One-sided congruence
Congruence lattice
Finitely generated congruence

Access to Document

10.1007/s10998-022-00497-zLicence: CC BY

Brookes_2022_PMH_Lattice-one-sided_CC
Copyright © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Final published version, 480 KBLicence: CC BY

Cite this

@article{3090489842a1406cb9c61f023ee1e629,

title = "The lattice of one-sided congruences on an inverse semigroup",

abstract = "We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of the trace and inverse kernel are discussed, in particular that the inverse kernel is a full inverse subsemigroup and that both the trace and inverse kernel maps are onto ∩-homomorphisms. It is shown that a left congruence is determined by its trace and inverse kernel, and the lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We demonstrate that every finitely generated left congruence is the join of a finitely generated trace minimal left congruence and a finitely generated idempotent separating left congruence. Characterisations are given of inverse semigroups that are left Noetherian, or are such that Rees left congruences are finitely generated.",

keywords = "Inverse semigroup, Inverse monoid, One-sided congruence, Congruence lattice, Finitely generated congruence",

author = "Matthew Brookes",

note = "Funding: This project forms part of the work toward my PhD at the University of York, supported by EPSRC grant EP/N509802/1.",

year = "2023",

month = sep,

day = "1",

doi = "10.1007/s10998-022-00497-z",

language = "English",

volume = "87",

pages = "1--26",

journal = "Periodica Mathematica Hungarica",

issn = "1588-2829",

publisher = "Springer",

number = "1",

}

TY - JOUR

T1 - The lattice of one-sided congruences on an inverse semigroup

AU - Brookes, Matthew

N1 - Funding: This project forms part of the work toward my PhD at the University of York, supported by EPSRC grant EP/N509802/1.

PY - 2023/9/1

Y1 - 2023/9/1

N2 - We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of the trace and inverse kernel are discussed, in particular that the inverse kernel is a full inverse subsemigroup and that both the trace and inverse kernel maps are onto ∩-homomorphisms. It is shown that a left congruence is determined by its trace and inverse kernel, and the lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We demonstrate that every finitely generated left congruence is the join of a finitely generated trace minimal left congruence and a finitely generated idempotent separating left congruence. Characterisations are given of inverse semigroups that are left Noetherian, or are such that Rees left congruences are finitely generated.

AB - We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of the trace and inverse kernel are discussed, in particular that the inverse kernel is a full inverse subsemigroup and that both the trace and inverse kernel maps are onto ∩-homomorphisms. It is shown that a left congruence is determined by its trace and inverse kernel, and the lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We demonstrate that every finitely generated left congruence is the join of a finitely generated trace minimal left congruence and a finitely generated idempotent separating left congruence. Characterisations are given of inverse semigroups that are left Noetherian, or are such that Rees left congruences are finitely generated.

KW - Inverse semigroup

KW - Inverse monoid

KW - One-sided congruence

KW - Congruence lattice

KW - Finitely generated congruence

U2 - 10.1007/s10998-022-00497-z

DO - 10.1007/s10998-022-00497-z

M3 - Article

SN - 1588-2829

VL - 87

SP - 1

EP - 26

JO - Periodica Mathematica Hungarica

JF - Periodica Mathematica Hungarica

IS - 1

ER -

The lattice of one-sided congruences on an inverse semigroup

Abstract

Keywords

Access to Document

Fingerprint

Cite this