What is the term for relation whose inversion is a function?

Question

Do we have a conventional term/name for such a relation $R$ (which is not necessarily a function) that $R^{-1}$ is a function?

If not, what are your suggestions?

Answer 1

Let $R$ be a relation in $A \times B$.

If you want $R^{-1}$ to be function on $B$, then you need $R$ to be injective and surjective.

Indeed, if $R$ is injective, then $R^{-1}$ is a function on the range of $R$.

If $R$ is surjective, then its range is $B$.

(A relation is injective iff $(a_1,b) \in R$ and $(a_2,b) \in R$ imply $a_1=a_2$.

A relation is surjective iff for every $b \in B$ there is $a \in A$ such that $(a,b) \in R$.)