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Given a partial function $f : A \rightarrow B$, and a subset $S \subseteq A$, we get a new partial function $$f \restriction_S : A \rightarrow B$$ by restriction. However, I prefer to analyse $f \restriction_S$ as a composite. To each subset $S$ of $A$, there's a corresponding a partial function $[S] : A \rightarrow A$ satisfying $\mathrm{supp}[S] = S$ and $S \subseteq \mathrm{id}_A$, that could be called the "restrictor" for $S$ or something like that, because for each partial function $f : A \rightarrow B$, we have: $f \restriction_S = f \circ [S].$ I much prefer the latter notation; for starters, we can put the restrictor on the other side to restrict at the codomain, as in $[S] \circ f$. It also makes it really obvious that $g \circ (f \restriction_S) = (g \circ f) \restriction_S,$ which just ends up being a special case of associativity. Also, to some mathematicians, the $\restriction$ notation indicates a change of domain, but in the partial function viewpoint, we usually don't wish to change domain, it's still just $A$.

Anyway, it would be nice to have standard notation for what I'm denoting $[S]$. Is there one?

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  • $\begingroup$ It seems to me that what you denote by $[S]$ can also be expressed as $\text{id}\upharpoonright S$. $\endgroup$
    – Joel David Hamkins
    Commented Apr 2, 2017 at 12:08
  • $\begingroup$ @JoelDavidHamkins, yep, they're inter-definable like that. But I'd rather just avoid the $\restriction$ notation altogether, so it seems a little self-defeating to define it that way, at least from my slightly idiosyncratic point of view. $\endgroup$
    – goblin GONE
    Commented Apr 2, 2017 at 12:16
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    $\begingroup$ Personally, I would find that perhaps the bracket notation $[S]$ might be overloaded, since this also often has other meanings, such as an equivalence class or the notation $[n]=\{1,2,\ldots,n\}$ that some people use. $\endgroup$
    – Joel David Hamkins
    Commented Apr 2, 2017 at 12:25
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    $\begingroup$ A partial relation is just a triple $(X, Y, r)$, for which $X$ and $Y$ are sets and $r \subseteq X \times Y$: In the practice of everyday life, we write $r: X \rightharpoonup Y$, or something similar, in place of the triple, and we take the domain of $r$ to be the set ${\rm dom}(r) := \{x \in X: (x,y) \in r\text{ for some }y \in Y\}$ (btw, wouldn't it be good to discourage the use of "domain" in cat theory as a synonymous of "source"?). Now, if you want an explicit reference to the domain, $D$, of $r$ in your notation, why not using something like $r: X \stackrel{D}{\rightharpoonup} Y$? $\endgroup$
    – Salvo Tringali
    Commented Apr 2, 2017 at 12:29
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    $\begingroup$ In semigroup theory it is common to use $1_S$ to denote the partial identity on S. You should Google restriction semigroups and categories where pele axiomatized this kind of stuff. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 2, 2017 at 16:56

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