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

Let TX be the full transformation monoid over a finite set X, and fix some a∈TX of rank r. The variant TXa has underlying set TX, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(TXa) consisting of all regular elements of TXa, and the lattice Cong(P) of all such congruences....

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Bibliographic Details
Published inJournal of algebra Vol. 662; pp. 431 - 464
Main Authors Dolinka, Igor, East, James, Ruškuc, Nik
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.01.2025
Subjects
Congruence
Congruence lattice
Full transformation semigroup
Subdirect product
Variant
Variant
Congruence
Subdirect product
08A30
20M10
20M20
Full transformation semigroup
Congruence lattice
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Summary:Let TX be the full transformation monoid over a finite set X, and fix some a∈TX of rank r. The variant TXa has underlying set TX, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(TXa) consisting of all regular elements of TXa, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of Cong(Tr), 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.
ISSN:0021-8693
DOI:10.1016/j.jalgebra.2024.08.017