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For instance for binary operators a Homomorphism is said to preserve mathematical structure of this kind, but for regular functions in set theory an isomorphism is used instead, and for topological spaces an homeomorphism is used.

Is there any book, mathematical subject or any resource that could help me understand why these definitions are given and not others and why they work in order to preserve mathematical structures, could other alternative definitions work as well?

Does model theory or category theory help to give further insight to understand this?

Thanks a lot