A natural partial order on the semigroups \Large S_n^- of order-decreasing transformations

Let T_X be the full transformation semigroup on a total order set X = \ 1 <2 < \cdots < n\ . Then S_n^ - = \left\ f \in T_X:\forall x \in X,f(x) \le x \right\ is a subsemigroup of T_X. We endow the order-decreasing transformation semigroup S_n^ - with the natural partial order. With respect to this partial order, we investigate when two elements of \Large S_n^- are related, then find elements of \Large S_n^- which are compatible with the order. Also, we characterize the minimal elements and the maximal elements of \Large S_n^- .