Variants of finite full transformation semigroups

TL;DR

This paper investigates the algebraic structure of variants of the full transformation semigroup on finite sets, focusing on their subsemigroups, ideals, ranks, and generating sets, providing detailed structural and combinatorial insights.

Contribution

It offers a comprehensive analysis of the structure, ranks, and generating sets of variants of the full transformation semigroup, including subsemigroups and ideals.

Findings

Calculated the rank and idempotent rank of the variants.

Determined the structure of regular elements and idempotent-generated subsemigroups.

Counted minimal generating sets for the variants.

Abstract

The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.