\begin{definition}[Definition:Ordered Subset]
Let $R = \struct{S, \preceq}$ be a relational structure.
{{Questionable|A relational substructure of a relational structure is not necessarily ordered. So either: (a) $\struct{S, \preceq}$ needs to be ordered; (b) $\struct{S', \preceq'}$ needs to be ordered; or (c) the page needs to be renamed. The theorem Ordered Subset of Ordered Set is Ordered Set and many links to the page suggests (a) is needed.|Leigh.Samphier}}
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
:$\forall x, y \in S': x \preceq y \iff x \preceq' y$
\end{definition}