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This is just an idea that I made up, and I have no idea if has been studied elsewhere (if it has, I'd appreciate links to some literature about it). It requires a bit of explanation because it's hard to think about (at least to me), so I'll begin with an analogy.

When we're dealing with a univariate function $f(x)$, we can just "take its derivative," but with a bivariate (or multivariate) function $f(x,y)$, it doesn't make sense just to "take a derivative" anymore. Instead, we take "partial derivatives," differentiating with respect to one variable and treating the other variable like a constant.

Now think about inverse functions. We can easily find the inverse of an invertible function $f(x)$, but we can't just "invert" a multivariate function like $f(x,y)$. Instead, I've defined a "partial inverse" analogously to a partial derivative. The partial inverse of a multivariate function $f(x_1,x_2,...,x_n)$ with respect to $x_i$ is denoted $$\text{inv}_{x_i}\space f(x_1,x_2,...,x_n)$$ and is found by treating $x_k$ with $k\ne i$ as constants and inverting $f$ as if it were simply a function of $x_i$ with some parameters. For example, if $g$ is defined as $$g(x,y)=x+2y+xy-1$$ Then, as per my definition, $$\text{inv}_x\space g(x,y)=\frac{x-2y+1}{y+1}$$ and $$\text{inv}_y\space g(x,y)=\frac{y-x+1}{x+2}$$ Of course, partial inverses of a function $f$ exist iff $f$ is a bijective function of $x$ whenever $y$ is fixed, and a bijective function of $y$ whenever $x$ is fixed. After playing with this idea for a while, I've come up with a few trivial properties of partial inverses, like $$f(\text{inv}_x\space f(x,y),y)=x$$ $$f(x,\text{inv}_y\space f(x,y))=y$$ However, the most interesting result I've come across is the following: $$\text{inv}_x\space\text{inv}_y\space\text{inv}_x\space f(x,y)=f(y,x)$$ One thing that I have explored without success and would like to know more about is the expression $$f(\text{inv}_x\space f(x,y),\text{inv}_y\space f(x,y))$$ So far, I've found no good way to simplify it.

QUESTION: Has this been studied before? If so, where? If not, can you guys let me know if you notice/find any interesting properties of partial inverses?

Do you have any suggestions for a better notation, or another suggestion for an extension of inverse functions to include multivariate functions?

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