 Mathematical name for groupBy

It seems many programming languages have a function similar to groupBy. Does anyone know whether there is a general mathematical name for this function? I.e., if f is a function (in the set theory sense) mapping set X to Y, f inverse may not exist, but I certainly can pair every element y in the range with the set of elements x, in the domain which f(x) = y.

If f maps X to Y, then groupBy(f) (for lack of better notation) maps Y to the power set of X, and is a well-defined function.

If I’m not mistaken, I believe such an operator also exists in relational algebra. Perhaps that name might be useful and evocative.

Does anyone know if this function has a name? I’m tempted to call it the reciprocal function.

I’ve posed the same question on math stack exchange.

If you interpret grouped values as equivalent, groupBy gives you the set of equivalence classes, also known as the quotient set.

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Yes, and my question is if f maps X to Y then what is the mathematical name for the function X.groupBy(f) called? I.e., X.groupBy(f) returns something which behaves like a function which can be called with a value from Y and it returns a subset of X. In scala it returns a List of elements from X, but mathematically/semantically it is a subset of X.

val strings = List("a","aa","abc","bb","c")
val g:Int=>List[String] = strings.groupBy(_.length)

g(2) // returns List("aa","bb")

From Wiki:

The surjective map x |-> [x] from X onto X / R , which maps each element to its equivalence class, is called the canonical surjection or the canonical projection map .

X / R is the quotient set. X is your space and R the relation

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Nice!