On the Monoid of All Order Preserving Full Contractions with a Fixed Set

In this paper, we consider the monoid $\text{Fix}\mathcal{O}\mathcal{C}\mathcal{T}\left(\left[n\right],D\right)$ of all order-preserving full contraction mappings that fix a subset say $D$ of a finite $n$ -element chain $\left\{1,2,？,n\right\}$ . We characterize regularity, Green’s relations, and starred Green’s relations, and show that this monoid is left adequate. Furthermore, we determine the cardinality of $\text{Fix}\mathcal{O}\mathcal{C}\mathcal{T}\left(\left[n\right],D\right)$ , the ranks of its two-sided ideals, and demonstrate that the ranks of the two-sided ideals and their corresponding Rees quotients are equal. Moreover, we deduce the rank of the monoid $\text{Fix}\mathcal{O}\mathcal{C}\mathcal{T}\left(\left[n\right],D\right)$ .