Definition:Restriction

Restriction of a Relation
Let $$\mathcal{R}$$ be a relation on $$S \times T$$.

Let $$X \subseteq S, \mathrm{Im} \left({X}\right) \subset Y \subseteq T$$.

The restriction of $$\mathcal{R}$$ to $$X \times Y$$ is defined as:

$$\mathcal{R} |_{X \times Y}: X \to Y = \mathcal{R} \cap X \times Y$$

If the range of $$\mathcal{R} |_{X \times Y}$$ is understood to be $$\mathrm{Rng} \left({\mathcal{R}}\right)$$, i.e. $$Y = \mathrm{Rng} \left({\mathcal{R}}\right)$$, then we define the restriction of $$\mathcal{R}$$ to $$X$$ as:

$$\mathcal{R} |_X: X \to \mathrm{Rng} \left({\mathcal{R}}\right) = \mathcal{R} \cap X \times \mathrm{Rng} \left({\mathcal{R}}\right)$$

An alternative way of saying the same thing is:

$$\mathcal{R} |_X = \left\{{\left({x, y}\right) \in \mathcal{R}: x \in X}\right\}$$

Restriction of a Mapping
Let $$f: S \to T$$ be a mapping.

Let $$X \subseteq S, \mathrm{Im} \left({X}\right) \subset Y \subseteq T$$.

The restriction of $$f$$ to $$X \times Y$$ is defined as:

$$f |_{X \times Y}: X \to Y = f \cap X \times Y$$

If the range of $$f |_{X \times Y}$$ is understood to be $$\mathrm{Rng} \left({f}\right)$$, i.e. $$Y = \mathrm{Rng} \left({f}\right)$$, then we define the restriction of $$f$$ to $$X$$ as:

$$f |_X: X \to \mathrm{Rng} \left({f}\right) = f \cap X \times \mathrm{Rng} \left({f}\right)$$

An alternative way of saying the same thing is:

$$f |_X = \left\{{\left({x, y}\right) \in f: x \in X}\right\}$$

This definition follows directly from that for a relation owing to the fact that a mapping is a special kind of relation.

Restriction of an Operation
In the same way that a restriction is defined on a relation, it can be defined on a binary operation.

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:

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

The notation $$\circ|_T$$ is generally used only if it is necessary to emphasise that $$\circ|_T$$ is strictly different from $$\circ$$ (through having a different domain and range). When no confusion is likely to result, $$\circ$$ is generally used for both.

Thus in this context, $$\left({T, \circ|_T}\right)$$ and $$\left({T, \circ}\right)$$ mean the same thing.

Obvious results:

If the operation $$\circ$$ is associative on $$\left({S, \circ}\right)$$, then it is also associative on $$\left({T, \circ}\right)$$:

$$T \subseteq S$$

$$\Longrightarrow \forall a, b, c \in T: a, b, c \in S$$

$$\Longrightarrow a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c$$

If the operation $$\circ$$ is commutative on $$\left({S, \circ}\right)$$, then it is also commutative on $$\left({T, \circ}\right)$$:

$$T \subseteq S$$

$$\Longrightarrow \forall a, b \in T: a, b \in S$$

$$\Longrightarrow a \circ b = b \circ a$$