  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.
