Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Igor Dolinka; James East; Nik Ruskuc

doi:10.1016/j.jalgebra.2024.08.017

Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Igor Dolinka^*, James East^*, Nik Ruskuc^*

^*Corresponding author for this work

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

Abstract

Let 𝒯_X be the full transformation monoid over a finite set X, and fix some a ∈ 𝒯_X of rank r. The variant 𝒯^a_X has underlying set 𝒯_X , and operation f * g = fag. We study the
congruences of the subsemigroup P = Reg(𝒯^a_X) consisting of all regular elements of 𝒯^a_X, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of
Cong(𝒯_r), and the full equivalence relation lattices of certain
combinatorial systems of subsets and partitions. We use this
to give an explicit classification of the congruences themselves,
and we also give a formula for the height of the lattice.

Original language	English
Pages (from-to)	431-464
Journal	Journal of Algebra
Volume	662
Early online date	9 Sept 2024
DOIs	https://doi.org/10.1016/j.jalgebra.2024.08.017
Publication status	Published - 15 Jan 2025

Keywords

Congruence
Congruence lattice
Full transformation semigroup
Variant
Subdirect product

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Dolinka-2024-Congruences-of-maximum-JAlgebra-662-431-CCBY
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title = "Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups",

abstract = "Let 𝒯X be the full transformation monoid over a finite set X, and fix some a ∈ 𝒯X of rank r. The variant 𝒯aX has underlying set 𝒯X , and operation f * g = fag. We study thecongruences of the subsemigroup P = Reg(𝒯aX) consisting of all regular elements of 𝒯aX, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product ofCong(𝒯r), and the full equivalence relation lattices of certaincombinatorial systems of subsets and partitions. We use thisto give an explicit classification of the congruences themselves,and we also give a formula for the height of the lattice.",

keywords = "Congruence, Congruence lattice, Full transformation semigroup, Variant, Subdirect product",

author = "Igor Dolinka and James East and Nik Ruskuc",

year = "2025",

month = jan,

day = "15",

doi = "10.1016/j.jalgebra.2024.08.017",

language = "English",

volume = "662",

pages = "431--464",

journal = "Journal of Algebra",

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TY - JOUR

T1 - Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

AU - Dolinka, Igor

AU - East, James

AU - Ruskuc, Nik

PY - 2025/1/15

Y1 - 2025/1/15

N2 - Let 𝒯X be the full transformation monoid over a finite set X, and fix some a ∈ 𝒯X of rank r. The variant 𝒯aX has underlying set 𝒯X , and operation f * g = fag. We study thecongruences of the subsemigroup P = Reg(𝒯aX) consisting of all regular elements of 𝒯aX, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product ofCong(𝒯r), and the full equivalence relation lattices of certaincombinatorial systems of subsets and partitions. We use thisto give an explicit classification of the congruences themselves,and we also give a formula for the height of the lattice.

AB - Let 𝒯X be the full transformation monoid over a finite set X, and fix some a ∈ 𝒯X of rank r. The variant 𝒯aX has underlying set 𝒯X , and operation f * g = fag. We study thecongruences of the subsemigroup P = Reg(𝒯aX) consisting of all regular elements of 𝒯aX, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product ofCong(𝒯r), and the full equivalence relation lattices of certaincombinatorial systems of subsets and partitions. We use thisto give an explicit classification of the congruences themselves,and we also give a formula for the height of the lattice.

KW - Congruence

KW - Congruence lattice

KW - Full transformation semigroup

KW - Variant

KW - Subdirect product

U2 - 10.1016/j.jalgebra.2024.08.017

DO - 10.1016/j.jalgebra.2024.08.017

M3 - Article

SN - 0021-8693

VL - 662

SP - 431

EP - 464

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Abstract

Keywords

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