Definition:Restriction of Ordering

Definition
Let $\struct {S, \preceq}$ 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:


 * ${\preceq \restriction_T} := {\preceq} \cap \paren {T \times T}$

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

Here, $\times$ denotes Cartesian product.

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

Also see

 * Restriction of Ordering is Ordering, which proves that $\preceq \restriction_T$ is an ordering on $T$

Technical Note
The expression:


 * ${\preceq \restriction_T} = {\preceq} \cap \paren {T \times T}$

is produced by the following $\LaTeX$ code:

{\preceq \restriction_T} = {\preceq} \cap \paren {T \times T}