# 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.

## 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.

## Also see

## 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.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 8$