Definition:Restriction/Operation
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Definition
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 codomain). 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 currently more likely to be seen 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.
Some authors prefer not to subscript the subset, and render the notation as:
- $f \restriction X = \left\{{\left({x, f \left({x}\right)}\right): x \in X}\right\}$
Also see
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 8$
