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

2024/03/08 by Igor Dolinka, James East, Dolinka, Igor +3
Computer Science · Decision Sciences · #Advanced Algebra and Logic #FOS: Mathematics #Fuzzy and Soft Set Theory #Group Theory (math.GR) #Rings and Algebras (math.RA) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2403.05191

openalex publication_date 2024/03/08 · openalex created_date 2024/03/13 · openalex updated_date 2026/07/28

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_X^a has underlying set T_X, and operation f⋆ g=fag. We study the congruences of the subsemigroup P=Reg(T_X^a) consisting of all regular elements of T_X^a, 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.

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