# Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition:

$$f(x,y) = x+y \\ g(y,z) = z-y \\ h(z,x) = z-x$$

Such collections can of course be generalized to any number of arguments.

Is there a name for $g$ and $h$? (It's not "partial inverse," though that would be a great name for them, analogous to "partial derivative.")

I believe they exist if $f$ is continuous, surjective, and monotone in each argument. Under what other conditions do they exist?

In general, what search terms should I use to find properties about them?

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Should this get moved to mathoverflow.net? – Neil Toronto Oct 21 '13 at 4:11
Asked at mathoverflow.net (mathoverflow.net/questions/145676/…) – Neil Toronto Oct 23 '13 at 21:23