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

Let \(\mathcal{CT}_n\) be the semigroup of full contraction mappings on \([n]=\{1,2,\ldots,n\}\), and let \(\mathcal{OCT}_n\) and \(\mathcal{ODCT}_n\) 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 \(\mathcal{ODCT}_n\) is left adequate. We further study the rank properties and as well obtain the rank of the semigroup, \(\mathcal{ODCT}_n\). Moreover, we obtain a characterization of natural partial order for the semigroup \(\mathcal{OCT}_n\) and its subsemigroup \(\mathcal{ODCT}_n\), respectively.