Is "is homomorphic to" an equivalence relation? (Hint:the difficulty is to decide on an appropriate meaning for the quoted phrase.)

Answer

Either yes (trivially) or no (nearly trivially).

If "is homomorphic to" is taken to mean there is a homomorphism from into (but not necessarily onto) , then every space is homomorphic to every other space as a zero map always exists.

If "is homomorphic to" is taken to mean there is an onto homomorphism from to then the relation is not an equivalence. For instance, there is an onto homomorphism from to (projection is one) but no homomorphism from onto , so the relation is not reflexive.

Can you help with showing where (in both cases) are the three properties of being symmetric, reflexive and transitive?

Thanking you in anticipation