Timeline for Is there standard notation for restriction partial functions?
Current License: CC BY-SA 3.0
15 events
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| Apr 2, 2017 at 19:23 | comment | added | Gerhard Paseman | I think the author is Peter Burmeister, who emphasizes the model theory aspect of partial algebras. You might consider his approach. Gerhard "Thinks It Was Peter Something-or-other" Paseman, 2017.04.02. | |
| Apr 2, 2017 at 18:52 | comment | added | Gerhard Paseman | I would make different suggestions depending on how you were to use it. In Universal Algebra, I might denote it as an additional collection of projection functions parametrized by S, say $p^i_S(x,y,...,z)$, especially if for clones of partial functions. (I forget the author's name of a relevant U.A. book, maybe Kreutzberger? Anyway, look at Graetzer's classic U.A. text too.) For Recursion Theory, I might emphasize the composition more. I don't know if Oddifreddi's text handles this, but that is where I would look first. Gerhard "And That's Just Two Perspectives" Paseman, 2017.04.02. | |
| Apr 2, 2017 at 16:56 | comment | added | Benjamin Steinberg | 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. | |
| Apr 2, 2017 at 14:06 | comment | added | Salvo Tringali | @goblin I'm afraid that we are getting far from the original topic of the thread (and it's my fault), and I don't want to give you the impression that I'm trying to convince you that my point of view is more correct than yours. But let me remark that I was not arguing against the use of the term "domain" in other fields than category theory. | |
| Apr 2, 2017 at 13:45 | comment | added | goblin GONE | @SalvoTringali, and I should add the codomain has a very entrenched meaning, and it's not synonymous with "range." | |
| Apr 2, 2017 at 13:44 | comment | added | goblin GONE | @SalvoTringali, but try using "source" in real life, and you'll begin to see the problem almost immediately. The source of $x^2$ is the real line. The source of the Lebesgue measure is the collection Lebesgue-measureable sets. It just sounds... weird. And that's assuming you're talking to a category theorist! If you're talking to a geometric measure theorist, they're going to start looking at you very strangely indeed. Just my opinion, of course. | |
| Apr 2, 2017 at 13:26 | comment | added | Salvo Tringali | @goblin Normally, I would just call them "relations": I'm writing of "partial relations" here to contrast them with "total relations", much in the same (redundant) way as we use the term "total function", rather than simply "function", to emphasize a difference with "partial functions", at least in some contexts. As for the rest, I'm not sure to understand what you mean with "sound weird and hard to understand", wrt the use of "domain" as a synonymous of "source" (which I find confusing, especially when it comes to "partial morphisms"): The latter is already widespread in CT. | |
| Apr 2, 2017 at 13:04 | comment | added | goblin GONE | @SalvoTringali, I don't like that terminology, because I know from experience that if you start replacing "domain" with "source" everywhere, you start to sound weird and hard to understand :). In the end, I decided to refer to the "domain" as you call it as the partial function's support. I'm also not quite sure why you refer to them as "partial relations". For me, a partial relation $X \rightarrow Y$ is the same thing as a relation $X \rightarrow 1+Y$. Such things are more fundamental and useful than most people realize, methinks. | |
| Apr 2, 2017 at 12:31 | comment | added | Noah Schweber | When in doubt, create your own notation and define it explicitly and precisely. This is certainly common practice. I don't know of any standard notation for $[S]$, but I would suggest $id_{S\subseteq A}$; and if you just want to avoid "$\upharpoonright$", you could always write "$f_{S\subseteq A}$" and (if you define it at the beginning of the paper) this would be pretty intuitive. (I agree with Joel though, the notation "$[S]$" is pretty overloaded if you use it this way.) | |
| Apr 2, 2017 at 12:29 | comment | added | Salvo Tringali | 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 $r: X \stackrel{D}{\rightharpoonup} Y$? | |
| Apr 2, 2017 at 12:29 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| Apr 2, 2017 at 12:25 | comment | added | Joel David Hamkins | 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. | |
| Apr 2, 2017 at 12:16 | comment | added | goblin GONE | @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. | |
| Apr 2, 2017 at 12:08 | comment | added | Joel David Hamkins | It seems to me that what you denote by $[S]$ can also be expressed as $\text{id}\upharpoonright S$. | |
| Apr 2, 2017 at 11:56 | history | asked | goblin GONE | CC BY-SA 3.0 |