Definition:Ordered Set of Increasing Mappings

Definition
Let $L = \left({S_1, \preceq_1}\right)$, $K = \left({S_2, \preceq_2}\right)$ be ordered sets.

The ordered set of increasing mappings from $K$ into $L$ is subset of $L^{S_2} = \left({S_1^{S_2}, \preceq'}\right)$ and is defined by
 * $\operatorname{Increasing}\left({K, L}\right) = \left({X, \precsim}\right)$

where
 * $X = \left\{ {f:S_2 \to S_1: f}\right.$ is increasing mapping$\left.\right\}$
 * $\mathord\precsim = \mathord\preceq' \cap \left({X \times X}\right)$
 * $L^{S_2}$ denotes the ordered set of all mappings from $S_2$ into $L$,
 * $S_1^{S_2}$ denotes the set of all mappings from $S_2$ into $S_1$.

$\operatorname{Increasing}\left({K, L}\right)$ as subset of ordered set is ordered set.