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:

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

If the range of $$\mathcal{R} \restriction_{X \times Y}$$ 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:

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

An alternative way of saying the same thing is:

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

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:

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

If the range of $$f \restriction_{X \times Y}$$ 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:

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

An alternative way of saying the same thing is:

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

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:

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

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 range). 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.

Notation
The use of the symbol $$\restriction$$ is a recent innovation over the more commonly-encountered $$|$$.

Thus the notation $$\mathcal{R} |_{X \times Y}$$ and $$\left({T, \circ|_T}\right)$$, etc. are more likely to be seen at the moment than $$\mathcal{R} \restriction_{X \times Y}$$ and $$\left({T, \circ\restriction_T}\right)$$.

No doubt as the convention becomes more established, $$\restriction$$ will develop.

It is strongly arguable that $$\restriction$$, affectionately known as "harpoon", is preferable to $$|$$ as the latter is suffering from the potential ambiguity of overuse.