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Let $f$ is a $n$-ary function (where $n$ is any index set) and let every argument of $f$ may be zero or non-zero (for example, we can consider arguments of $n$ being posets with least element which I call "zero").

Are there any special name for such $f$ that every argument can be restored knowing the value of the function, provided that every of the $n$ arguments is non-zero?

An example of such $f$ is cartesian product of $n$ sets (where zero is the empty set).

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The general terminology is that $f$ is invertible. For your specific example where $f$ is only invertible when all arguments are non-zero, then, assuming each argument of $f$ is in $\mathbb{C}$, this can be described by saying that $f|_{\mathbb{C}^{*n}}$ (the restriction of $f$ to $\mathbb{C}^{*n}$) is invertible ($\mathbb{C}^*$ being all nonzero elements of $\mathbb{C}$).

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