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

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.