I recently encountered what seems to me like an inconsistency in the usage of the term bijective. This is pretty basic stuff, but it somehow never occurred to me before. I'd like to make sure I'm using the term properly with respect to relations.
The definition (and, I assume, etymology) of bijective is usually given for functions as:
(1) injective and surjective.
But I've also heard the term used for relations. Thinking back, it would refer to a relation that is:
(2) injective and uniquely defined.
The point being that the bijective property should actually refer to the "one-to-one" nature of the relation or function in question.
(Functions get uniquely defined 'for free'. The extra ingredient for a bijective function is surjectivity, probably with the purpose that its inverse is then also a bijective function.)
But the two definitions are incompatible with the interpretation of functions as well-defined relations. Taking them both literally would result in contradiction: a non-surjective function would be bijective by (2), but not bijective by (1).
How is this usually dealt with? Is there a generally accepted alternative term for (2)?
Edit 1: As it turns out I am unable to find any reference to definition (2). I suppose my new question now becomes: Is there a nice term for uniquely defined, injective relations / injective partial functions?
Edit 2: Take a look at this Wikipedia section:
https://en.wikipedia.org/wiki/Finitary_relation#Analogy_with_functions
According to this, a relation can be bijective without being a function. It still doesn't correspond to (2) though. It keeps its etymological meaning: injective and surjective, i.e., the inverse of a function.
The nouns bijection, injection and surjection are reserved for functions, but the adjectives can apply to relations too.
(2) is called one-to-one there which, I suppose, is acceptable. 🙂
Best Answer
Actually "bijective", like "injective" and "surjective", is a perfectly well defined notion, and any object that claims this badge has to be a function. Most people would not consider applying the notion at all in a context where the object is not assumed to be a function to begin with, but it is acceptable to define a bijective relation to be a one-to-one correspondence, in other words a relation that is actually a bijective function (this seems to be the case for this definition, which does not correspond to (2) of the question).
Other notional may be less clearly defined; notably "one-to-one" has always been a mystery to me, because it means "bijective" when used as in my previous sentence, but a "one-to-one map" is actually only an injective function.
The notion descibed in (2) might be called an "injective partial function" as you do without causing much confusion. However it does create the precedent of applying the adjective "injective" to something that is not a function (or a module, a resolution, or a metric space). Alternative terms one could think of proposing are "invertible partial function" (but it is not really invertible, as composition with its "inverse" partial function only gives a partial identity), or "partial bijection" (but one has to understand that "partial" applies to both sides, so nothing of surjectivity is left) or "partial injection" (no confusion, but like "injective partial function" the terminology appears to be asymmetric, while the notion itself is not). Finally "zero-or-one-to-zero-or-one correspondence" does seem to suggest the proper definition, but is frankly quite awkward.