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A mapping that maps at most one input to a given output is called injective (adjective). It has injectivity.

A mapping that maps a given input to at most one output is called a function (noun). It has ... functionality?

Is there clearer phrasing for this? I'm writing a computer science thesis, and one of my hypotheses states that it is a problem that a certain transformation maps to at most one output given an input, but obviously it's ambiguous to write about how "functionality" is an issue. Also, a functional is a thing by itself.

Edit: I forgot to mention that my mapping can be interpreted as a "predecessor" relation in a directed graph. As it stands, all the nodes in the graph have either $0$ predecessors or $1$ "left" and $1$ "right" predecessor. There are reasons why this is problematic, and that actually, there should be $n>1$ left and right predecessor pairs. Hence, a mapping that would map a node as input onto its left predecessors is currently a function, but it should map to a set of more than $1$ outputs (the amount can differ), no longer making it a function. Idem for the right-predecessor function.

Mew
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  • I assume you are using map the way "relation" is normally used..., You could say that the opposite relation, or inverse relation, is injective, but injectivity is used for functions and you aren't guaranteed that the inverse relation is a function, so I'm not sure if it really is less ambiguous – Carlyle May 04 '23 at 22:03
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    I don't think there is a standard term for such a thing, at least not one that's universally recognized. You can define whatever notion you want, however, in the context of what you are working on, or more generally if that makes sense. – lulu May 04 '23 at 22:03
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    @Carlyle Sure, yes. It's been a few years since I had abstract algebra :) – Mew May 04 '23 at 22:04
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    There isn't a word for this, as far as I know. Definitely, functionality is not it. – Thomas Andrews May 04 '23 at 22:08
  • @Mew sorry I accidentally sent halfway through the comment xD – Carlyle May 04 '23 at 22:08
  • I have added an edit that might reframe the conversation to graph theory instead of functional theory. – Mew May 04 '23 at 22:19
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    I think "functionality" is a sufficiently funny term that you should define it to mean this and then use it. – Misha Lavrov May 04 '23 at 22:23
  • It sounds like you're saying that your function is "well defined". – JonathanZ May 04 '23 at 22:47
  • @JonathanZsupportsMonicaC That does seem to match what I am saying (according to this thread), but sadly, well-defined is -- ironically -- an ambiguous word too. "It is an issue that this transformation is well-defined" makes me sound like a crazy person. – Mew May 04 '23 at 22:53
  • This is confusing. Why not define a map "lpred" that maps a node to a set of nodes, i.e. the set of all left predecessors, and then discuss whether on not it maps to singletons? If you're going to call something a function, it should be a function, and working towards it not being well defined goes against the conventional use of function language. – JonathanZ May 04 '23 at 23:03
  • @JonathanZsupportsMonicaC Two things: 1. the thesis critiques an assumption in literature that it should be a function. 2. Indeed, my whole point is that it should not be seen as a function. I can redefine it afterwards by generalising it (where the current understanding is a map to singletons), but I'd first like to stick a name onto the problem I'm seeing in the literature, which is in fact the assumption of "functionality" (but with a better word). – Mew May 04 '23 at 23:19
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    You should reproduce one of the definitions given for it in the literature. If it's something like "We define the sister() function as "sister(x) is the female sibling of x", then saying "this is not well-defined" is exactly how one would criticize it with mathematical language. – JonathanZ May 04 '23 at 23:34
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    @Mew Not an adjective, but a relation that happens to also be a function could be called a functional relation 1, 2. – dxiv May 05 '23 at 04:25

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