On certain Semigroup of Order-decreasing Full Contraction Mappings of a Finite Chain

Abstract

Let ${\mathcal{C}\mathcal{T}}_n$ be the semigroup of full contraction mappings on $[n]=\{1,2,\dots ,n\}$, and let ${\mathcal{O}\mathcal{C}\mathcal{T}}_n$ and ${\mathcal{O}\mathcal{D}\mathcal{C}\mathcal{T}}_n$ be the subsemigroups consisting of all order-preserving full contraction and subsemigroup of order-decreasing and order-preserving full contraction mappings, respectively. In this paper, we show that the semigroup ${\mathcal{O}\mathcal{D}\mathcal{C}\mathcal{T}}_n$ is left adequate. We further study the rank properties and as well obtain the rank of the semigroup, ${\mathcal{O}\mathcal{D}\mathcal{C}\mathcal{T}}_n$. Moreover, we obtain a characterization of natural partial order for the semigroup ${\mathcal{O}\mathcal{C}\mathcal{T}}_n$ and its subsemigroup ${\mathcal{O}\mathcal{D}\mathcal{C}\mathcal{T}}_n$, respectively.