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arXiv · 2403.05191

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

Igor Dolinka·James East·Nik Ruškuc
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Abstract

Let $T_X$ be the full transformation monoid over a finite set $X$, and fix some $a\in T_X$ of rank $r$. The variant $T_X^a$ has underlying set $T_X$, and operation $f\star g=fag$. 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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Igor Dolinka, James East, Nik Ruškuc. 2024-03-08. Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups. https://arxiv.org/abs/2403.05191

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