Definition:Restriction/Operation

Definition
Let $\struct {S, \circ}$ be an algebraic structure, and let $T \subseteq S$.

The restriction of $\circ$ to $T \times T$ is denoted $\circ {\restriction_T}$, and is defined as:


 * $\forall t_1, t_2 \in T: t_1 \mathbin {\circ {\restriction_T} } t_2 = t_1 \circ t_2$

The notation $\circ {\restriction_T}$ is generally used only if it is necessary to emphasise that $\circ {\restriction_T}$ is strictly different from $\circ$ (through having a different domain).

When no confusion is likely to result, $\circ$ is generally used for both.

Thus, in this context, $\struct {T, \circ {\restriction_T} }$ and $\struct {T, \circ}$ mean the same thing.

Also see

 * Definition:Extension of Operation


 * Definition:Restriction of Relation
 * Definition:Restriction of Mapping