Homomorphisms as "structure-preserving" maps A function f between groups is said to be a homomorphism if it "preserves" the product in the sense that f(xy)=f(x)f(y). A function f between fields is said to be a homomorphism if it "preserves" both addition and multiplication in the sense that f(xy)=f(x)f(y), f(x+y)=f(x)+f(y). Homomorphisms are often described as "structure-preserving maps". We should be able to define homomorphisms as structure-preserving maps, if we define "structure" first, and also what it means for the structure to be "preserved". I suggest the following definition: We define a structure as a 4-tuple (S,R,O,C), where S is a set, R is a set of relations on S, O is a set of operations on S, and C is a subset of S. The members of C are called constants. If (S,R,O,C) and (S',R',O',C') are structures, a map f:S→S' is said to be structure preserving if the following holds for all x,y,x1,...,xn in S: a) If r is a relation in R such that (x,y) is in r, then there's a r' in R' such that (f(x),f(y)) is in r'. b) For each n=1,2,..., if m is an operation of arity n in O, then f(m(x1,...,xn))=m(f(x1),...,f(n)) c) If x is in C, then f(x) is in C'. (A relation of arity n is a subset of Sn. An operation of arity n is a function from Sn into S). Unless I missed something essential (and I might have), this should take care of the definition of "homomorphism" for groups, rings, fields, topological spaces and a lot more, but not for metric spaces, vector spaces, manifolds and a lot more. A metric space involves a function that isn't an operation. A vector space also involves a function (scalar multiplication) that isn't an operation, and it also involves a previously defined structure (a field). And manifolds...uh...I don't even want to think about it. What I'd like to know if there is a more general definition that covers these other things as well. I know the term homomorphism is used for vector spaces, but I don't think I've heard it used for metric spaces. I've heard people say that diffeomorphisms are isomorphisms between manifolds, but perhaps they just meant isomorphisms as they are defined in category theory (where they don't have to be structure preserving). By the way, I have studied some mathematical logic since the other discussion, so I'm now familiar with how structures/algebras are defined in model theory/universal algebra. I understand the reason why texts on those subjects talk about relation symbols instead of relations and so on, and I feel that we don't need to do that here, as long as we're just looking for a single definition of homomorphism.