Say I had a function $f:\mathbb{R}_x\rightarrow\mathbb{R}_z$ with $z=f(x)=x+1$ (the subscripts on the $\mathbb{R}$ are just there to indicate which variable corresponds to which space). Since $f$ is bijective, I could define the inverse as $f^{-1}:\mathbb{R}_z\rightarrow\mathbb{R}_x$ with $x=f^{-1}(z)=z-1$. The key property of an inverse is that $\forall x \in \mathbb{R}_x\,,\ f^{-1}(f(x))=x$.
Now suppose I had a function $f:\mathbb{R}_x\times\mathbb{R}_y\rightarrow\mathbb{R}_z$ with $z=f(x,y)=x+y$. I can't really have an inverse because the map is not bijective, e.g. both $f(1,0)$ and $f(0,1)$ map to $1$.
However, for a project I'm working on, it would be useful to talk about a "partial" or "weak" inverse. For instance, with just $z$ I can't recover any inputs, but with both $z$ and $x$ I could recover $y$. So I could think of a partial/weak inverse as a function $f^{-1}_{\mathbb{R}_x}:\mathbb{R}_x\times\mathbb{R}_z\rightarrow\mathbb{R}_y$ with $y=f^{-1}_{\mathbb{R}_x}(z,x) = z-x$. This function is bijective, and $\forall x \in \mathbb{R}_x$ and $\forall y \in \mathbb{R}_y$, $f^{-1}_{\mathbb{R}_x}(f(x,y),x)=(x+y)-x=y$, which is not exactly the inverse property, but useful (for us at least) nonetheless.
Is there an existing name and notation for this kind of partial/weak inverse? Or is it something we'll just have to make up and define?
I already asked this question here: What mathematical term for this kind of "solution functions"?
If we don't have other terms/words, we can always use set theory:
Your function $f\colon (x,y)\mapsto z=x+y$ is a relation $R$ with $(x,y)Rz\Leftrightarrow z=x+y$.
The $y=z-x$ are the values of the projection function $\pi_2\colon (x,y,z)\mapsto y$.
Your $(x,z)\mapsto y$ is a relation $R'\colon (x,z)R'y\Leftrightarrow y=z-x$. See the different properties, types and terms of relations: Wikipedia German: Relation (Mathematik) - Eigenschaften zweistelliger Relationen. (The inverse (inverse function) exists only for bijective functions. But the inverse relation always exists. The inverse relation of a non-bijective function can be e.g. a multifunction (correspondence), a partial function or simply only a relation.)
$R$ and $R'$ are interrelated by the two functions $R\to R', ((x,y),z)\mapsto((x,z),y)$ and $R'\to R, ((x,z),y)\mapsto((x,y),z)$.
The functions $(x,y,z)\mapsto(x,z,y)$ and $(x,z,y)\mapsto(x,y,z)$ are permutations, they are bijective functions.
The term "partial inverse" seems to have already a different meaning: Wikipedia: Inverse function - Partial inverses. (The inverse relation can be decomposed into its functional branches, the partial inverses.)