Speaking in the strictest sense, does a homomorphism map between the carriers of two algebras, or between the two algebraic structures? To explicate what I mean suppose we have two algebras (j, J) and (k, K), which have carriers (sets) j and k respectively, and have binary operations J and K. Does a homomorphism H come as a function H:j->k (maps the carriers) such that for all x, y in g, H (J (x, y))=K ( H(x), H(y)), or does it come as a function H:(j, J)->(k, K)? Or do the two definitions come as equivalent in some sense? Say we have the following two structures:
A 1 2 1 1 2 2 2 2 B a b c a a b b b b b b c b c c
and the homomorphism L:{1, 2}->{a, b, c}, specified by L:1->a, 2->b. Also, suppose we have the same sets, but now with tables as follows:
A 1 2 1 1 2 2 2 2 B a b c a a b c b b b b c b c c
And say we have function M:{1, 2}->{a, b, c}, M:1->a, 2->b. Does function M come as different than function L above, since they map between different structured sets (if they do such, of course), or does L basically come as the same function as M since the have the same domain, the same co-domain, and the same set of ordered pairs (1, a), (2, b)?