I came across a problem where I don't know what field to consult. The problem is simple to explain:
I have a geometric object $S$, and a finite set of maps $V$ from the points in this object to $V$.
Intuitively, any $v \in V$ partitions $S$, and chosing one of these partitions yields a new such partitioning.
What kind of thing is this and/or what could I google to get more information on it or similar structures? (I am particularly interested in iterated applications, given an initial element of $V$ (partitioning), to a given sequence of points.)
Thank you in advance.