Definition:Ordered Subset

Definition
Let $R = \struct{S, \preceq}$ be a relational structure.

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

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