I'm looking for a very general way to specify a space of (abstract) locations structured in some arbitrary way. I do not want to have to impose any particular spatial/temporal structure on that set. Rather my goal is to specify a general way in which sets of locations in some space may be classified according to their structural properties. In particular, I want to classify sets of locations using equivalence classes (or disjunctive collections of equivalence classes) as types. The equivalence classes would be derived from the set of invariants generated by L and S (so that it covers the space). It should be a transformation group.
For example, suppose that I take each square on a chessboard to be a location. Then intuitively some sets of squares look like other sets of squares. In particular there is some kind structural invariant between them.
I'm looking at something like:
Let's say that L is a set of abstract locations and that S is a set of n-ary relations on L, called the structural relations. Say that any two sets of locations $A \subseteq L$ and $B \subseteq L$ are isomorphic if and only if there is a bijective function f:A$\to$B such that for all locations l in L and for each n-ary relation R in S, $ l_1,...,l_n$ is in R iff $f(l_1),...,f(l_n)$ is in R. The function f is an invariant.
Am I on the right track?
I'm grateful for any assistance anyone might offer.