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 defined as:


 * $\left({T, \circ \restriction_T}\right): t_1, t_2 \in T: t_1 \circ \restriction_T \ t_2 = t_1 \circ t_2$

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 and codomain). 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