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What is the conventional name for functions that compute the intersection of two other functions?

I have a function called I that computes the intersection of two functions, and also a function called F that computes the leftmost point of a function. For example, it may be used as I[F[x], F[y], F[z]] where the inputs are x, y and z.
I have a function that creates the leftmost point of a function, which I did not name G. Is there a commonly used nomenclature for such functions, which take two functions as inputs, and return a function?
(In the example, I omit the syntax of the functions for clarity, but the problem is real.)

A:

We sometimes use the notation$$\operatorname{IF}_{a,b}[g,h]$$to indicate that we are taking the intersection of $g$ and $h$ as functions of $a$ and $b$ (rather than to indicate that we are taking the intersection as a function of $a$ and $b$ as we use the more conventional $\operatorname{IF}$ notation for intersection of sets, depending on context).
With the example $$\operatorname{IF}_{F[x],F[y],F[z]}[g,h]$$which takes the intersection of $g$ and $h$ as functions of $F[x]$ and $F[y]$, and you could use (for example)$$\operatorname{IF}_{g,

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