This may be further afield: maps which preserve or promote some relation, while "encoding" the basic operation. I dimly recall groupoid self-maps called cryptomorphisms, which had some property like f(a*b)= p(f(a))*q(f(b)) for p and q some permutations on the set. Also, in classifying Latin squares, there is some notion of isotopy and some other relations that allow a quasigroup table to be related to others through certain maps.
There are similar examples where one is concerned with preserving some property like essential arity (e.g. binary operations which depend on both variables), and will insist on maps between structures that share or preserve such a property. Unfortunately all I remember at the moment is some variant of Mazurkiewicz, who along with similarly named people (speaking as an ignorant American) used variants of morphism to help study groupoids and other structures, partly to understand their spectra (number of algebras/operations/essential operations on an underlying set of n elements, n varying over finite numbers).
As an example of something I haven't seen but can imagine: an inequality map, where you care that f(A*B) <> f(C*D) if A*B <> C*D .
I invite others to add to this post as details occur and memories sharpen.
Gerhard "Ask Me About System Design" Paseman, 2010.02.09
