Definition:Restriction of Ordering
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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}