Definition:Ordered Subset

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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

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


Sources