Definition:Restriction/Operation

Definition
Let $\left({S, \circ}\right)$ 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, $\left({T, \circ {\restriction_T}}\right)$ and $\left({T, \circ}\right)$ mean the same thing.

Also see

 * Extension of an Operation


 * Restriction of a Relation
 * Restriction of a Mapping

Technical Note
The expression:


 * $t_1 \mathbin{\circ {\restriction_T}} t_2$

is produced by the following $\LaTeX$ code:

t_1 \mathbin{\circ {\restriction_T}} t_2