Definition:Restriction/Operation
Definition
Let $\struct {S, \circ}$ be an algebraic structure.
Let $A, B \subseteq S$.
The restriction of $\circ$ to $A \times B$ is denoted $\circ {\restriction_{A \times B} }$, and is defined as:
- $\forall a \in A, b \in B: a \mathbin {\circ {\restriction_{A \times B} } } b = a \circ b$
The notation $\circ {\restriction_{A \times B} }$ is generally used only if it is necessary to emphasise that $\circ {\restriction_{A \times B} }$ is strictly different from $\circ$ (through having a different domain).
When no confusion is likely to result, $\circ$ is generally used for both.
Notation
The use of the symbol $\restriction$ is a recent innovation over the more commonly-encountered $\vert$.
Thus the notation $\RR \vert_{X \times Y}$ and $\struct {T, \circ \vert_T}$, etc. are currently more likely to be seen than $\RR {\restriction_{X \times Y} }$ 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 $\vert$ 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 note that it is commonplace even to omit the $\restriction$ symbol altogether, and merely render as $\RR_{X \times Y}$ or $\struct {T, \circ_T}$, and so on.
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 \(s \mathrel {\RR {\restriction_{X \times Y} } } t\) is s \mathrel {\RR {\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): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets