Given two sets with some structure
imposed on it - for instance, a relation
, or an operation
, or an algebra
a homomorphism is a mapping from one to the other that preserve
s that structure.
E.g., the natural numbers can be mapped to the even natural numbers
by the mapping x -> 2x; this is a homomorphism with respect to the usual ordering of numbers.
x -> 2^x is a homomorphism to map addition onto multiplication.
Isomorphisms are the most common.