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Maybe this property could be called "exclusivity" ?

Does it have a standard name?

It recalls the definition of a function as a " single-valued relation" (Enderton).

But here, it is not required that any a ( in a given set) be related to some b.

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    $\begingroup$ It is referred to as being "Well-defined." Compare this to that every $a$ is related to some $b$ which is referred to by the phrase "Everywhere-defined." $\endgroup$ – JMoravitz May 22 at 13:43
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    $\begingroup$ @JMoravitz I've never heard "well-defined" used in this sense. Whenever I've heard "the relation $R$ is well-defined," what's been meant is "the definition of $R$ we gave was in terms of representations of objects, but in fact different representations of the same objects don't yield different results." $\endgroup$ – Noah Schweber May 22 at 13:46
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    $\begingroup$ @JMoravitz I see "Well-defined" as more of a term about functions. Plus, I would also expect a well-defined function to be defined on its domain. In the context of relations, calling this "well-defined" suggests that other relations (that are defined perfectly well) are not well-defined. $\endgroup$ – Theo Bendit May 22 at 13:46
  • $\begingroup$ I don't know of a name, but I'd consider calling such relations "many-to-one" or "codomain-exclusive". $\endgroup$ – Theo Bendit May 22 at 13:51
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Such relations are (in my experience) called "functional", in analogy with functions. Indeed, such a relation is a partial function (and actually I've heard "$R$ is a partial function" more frequently than I've heard "$R$ is functional").

Similarly, relations such that for every $a$ there is at least one $b$ with $aRb$ are called "total" (in analogy with partial vs. total functions), or "serial" (although I've heard that one much more rarely). And relations such that for each $b$ there is at most one $a$ with $aRb$ are called "injective" (or "one-to-one") relations.

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  • $\begingroup$ I'm curious, why the downvote? $\endgroup$ – Noah Schweber May 22 at 13:48
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    $\begingroup$ Injective would mean here that if $aRb$ that there is no $c$ different than $a$ such that $cRb$. This is a different condition than if $aRb$ that there is no $c$ different than $b$ such that $aRc$. Every function satisfies the intended condition. Only some functions satisfy the condition of being injective. $\endgroup$ – JMoravitz May 22 at 13:49
  • $\begingroup$ @NoahSchweber. Thanks for your answer. +1 $\endgroup$ – Eleonore Saint James May 22 at 13:49
  • $\begingroup$ @JMoravitz Derp, I'm an idiot - fixing! (I'm going to blame that one on math elves.) $\endgroup$ – Noah Schweber May 22 at 13:50
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    $\begingroup$ @EleonoreSaintJames See JMoravitz' comment (although I've fixed the issue now). $\endgroup$ – Noah Schweber May 22 at 13:52
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According to Encyclopedia of Math and nLab, such a relation is called a functional relation on a set.

A functional relation defines a partial function from the set to itself, so you might as well call it a partial function.

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