Definition:Restriction of Ordering

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$ be a subset of $S$.

Then the restriction of $\preceq$ to $T$, denoted $\preceq \restriction_T$, is defined as:


 * $\preceq \restriction_T := \left({T \times T}\right) \, \cap \preceq$

viewing $\preceq \subseteq S \times S$ as a relation on $S$.

Here, $\times$ denotes Cartesian product.

Thence restriction of $\preceq$ to $T$ is an instance of restriction of a relation.

Also see

 * Restriction of Ordering is Ordering, proving the name restricted ordering appropriate