Definition:Ordered Subset

Definition
Let $R = \left({S, \preceq}\right)$ be a relational structure.

A ordered subset of $R$ is a relational structure $\left({S', \preceq'}\right)$ such that
 * $S'$ is a subset of $S$,
 * $\mathord\preceq' = \mathord\preceq \cap \left({S' \times S'}\right)$

That is
 * $\forall x, y \in S': x \preceq y \iff x \preceq' y$