It depends what the fundamental operations are in your context. There are different types of map that we call "X homomorphism" for different types of algebraic

It depends what the fundamental operations are in your context. There are different types of map that we call "X homomorphism" for different types of algebraic structure X (groups, rings, fields, vector spaces, modules, algebras, etc). In an abstract algebra course you'd probably first encounter the definition of a 'group homomorphism' (preserves the group's multiplication operation, whatever that is: f(x*y) = f(x)*f(y)), and then a separate definition of a 'ring homomorphism' (preserves the ring's addition and multiplication, which are the two operations you have on a ring - f(x+y)=f(x)+f(y) and f(xy) = f(x)f(y)), etc. The definitions are similar, but they are distinct: a group homomorphism is between groups, a ring homomorphism is between rings, etc. (There's also a more formal sense in which they're examples of the same thing, you'd look into category theory for more on that.) So what's going on in your example? Well, you could define a simple type of algebraic structure where, for any instances of that structure, you can only ask the question "is x positive" for one of its elements x and get a yes or no answer, and then define 'homomorphisms' between such structures as maps that take 'positive' elements of the source space to 'positive' elements of the destination space (and vice versa). But I'm not sure you'd really be able to use it for much. (On https://en.wikipedia.org/wiki/Homomorphism#Definition https://en.wikipedia.org/wiki/Homomorphism#Definition, that'd correspond to having exactly one unary operation μ(x) = is_positive(x) and no other operations, not even addition.)