Definition:Restriction

Definition

Restriction of a Relation

Let $\mathcal R$ be a relation on $S \times T$.

Let $X \subseteq S$, $Y \subseteq T$.

The restriction of $\mathcal R$ to $X \times Y$ is the relation on $X \times Y$ defined as:

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

If $Y = T$, then we simply call this the restriction of $\mathcal R$ to $X$, and denote it as $\mathcal R {\restriction_X}$.

Restriction of a Mapping

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

Let $X \subseteq S$.

Let $f \sqbrk X \subseteq Y \subseteq T$.

The restriction of $f$ to $X \times Y$ is the mapping $f {\restriction_{X \times Y}}: X \to Y$ defined as:

$f {\restriction_{X \times Y}} = f \cap \paren {X \times Y}$

If $Y = T$, then we simply call this the restriction of $f$ to $X$, and denote it as $f {\restriction_X}$.

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 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.

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 $\struct {T, \circ|_T}$, etc. are currently more likely to be seen than $\mathcal R {\restriction_{X \times Y} }$ and $\struct {T, \circ {\restriction_T} }$.

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

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

Some authors prefer not to subscript the subset, and render the notation as:

$f \mathbin \restriction X = \set {\tuple {x, \map f x}: x \in X}$

but this is not recommended on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it has less clarity.

Technical Note

The $\LaTeX$ code for $f {\restriction_{X \times Y} }: X \to Y$ is f {\restriction_{X \times Y} }: X \to Y .

Note that because of the way MathJax renders the image, the restriction symbol and its subscript \restriction_T need to be enclosed within braces { ... } in order for the spacing to be correct.

The $\LaTeX$ code for $s \mathrel {\mathcal R {\restriction_{X \times Y} } } t$ is s \mathrel {\mathcal R {\restriction_{X \times Y} } } t .

The $\LaTeX$ code for $t_1 \mathbin {\circ {\restriction_T} } t_2$ is t_1 \mathbin {\circ {\restriction_T} } t_2 .

Again, note the use of \mathrel { ... } and \mathbin { ... } so as to render the spacing evenly.