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

Abstract

Let $[n]=\{1,2,\dots ,n\}$ be a finite chain and let ${\mathcal{C}\mathcal{T}}_n$ be the semigroup of full contractions on $[n]$. Denote ${\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{T}}_n$ and ${\mathcal{O}\mathcal{C}\mathcal{T}}_n$ 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$({\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{T}}_n)$ and Reg$({\mathcal{O}\mathcal{C}\mathcal{T}}_n$), respectively) and the collection of all idempotent elements (denoted by E$({\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{T}}_n)$ and E$({\mathcal{O}\mathcal{C}\mathcal{T}}_n$), respectively) of the subsemigroups ${\mathcal{O}\mathcal{R}\mathcal{C}\mathcal{T}}_n$ and ${\mathcal{O}\mathcal{C}\mathcal{T}}_n$, respectively are subsemigroups. In this paper, we study some combinatorial and rank properties of these subsemigroups.

10 pages