A relation $R$ may have the property that each element $y$ of $\mathrm{ran}\, R$ is the unique right value of $R$ at some object, that is, for some object $x$, $\langle x,z\rangle \in R$ if and only if $z = y$. Is there a recognized name for this property?

One reason that there might be is that it is a necessary and sufficient condition for the direct image function $R^{\rightarrow}\, \colon\, \mathcal{P}(\mathrm{dom}\, R) \to \mathcal{P}(\mathrm{ran}\, R)$ to be a surjection; where, for each member $A$ of $\mathcal{P}(\mathrm{dom}\, R)$, $R^{\rightarrow}(A)$ is the direct image of $A$ under $R$, that is, the set of objects $y$ such that, for some member $x$ of $A$, $\langle x, y \rangle \in R$.

  • $\begingroup$ Are you just looking for "is a function" (or some equivalent thereof), or am I misreading your question? $\endgroup$
    – tabstop
    Apr 3 at 1:47
  • $\begingroup$ You're misreading the question. The property is strictly weaker than that of being a function. For example, {<0,0>, <1,0>,<1,2>,<2,2>} satisfies it but is not a function. $\endgroup$
    – A. Burrell
    Apr 3 at 15:20


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