On the combinatorial and rank properties of certain subsemigroups of full contractions of a finite chain

TL;DR

This paper investigates the combinatorial structures and rank properties of certain subsemigroups of full contractions on finite chains, focusing on order-preserving and reversing contractions, and their regular and idempotent elements.

Contribution

It provides new insights into the combinatorial and rank characteristics of subsemigroups of full contractions, including those of regular and idempotent elements, extending prior work.

Findings

Characterization of regular elements in subsemigroups

Determination of rank properties of these subsemigroups

Analysis of combinatorial structures within the semigroups

Abstract

Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{CT}_{n}$ be the semigroup of full contractions on $[n]$. Denote $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$ to be the subsemigroup of order preserving or reversing and the subsemigroup of order preserving full contractions, respectively. It was shown in [17] that the collection of all regular elements (denoted by, Reg$(\mathcal{ORCT}_{n})$ and Reg$(\mathcal{OCT}_{n}$), respectively) and the collection of all idempotent elements (denoted by E$(\mathcal{ORCT}_{n})$ and E$(\mathcal{OCT}_{n}$), respectively) of the subsemigroups $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$, respectively are subsemigroups. In this paper, we study some combinatorial and rank properties of these subsemigroups.