Question:

The definition of a monotone map reminded me of homomorphism.

I remember seeing homomorphism in a data-base class, where it was defined as:

A homomorphism from a data-base instance \\( K1 \\) to another instance \\( K2 \\) is a mapping \\(h\\) from the domain of \\(K1\\) to the domain of \\( K2 \\) such that for every fact \\( R(a_1, ..., a_n) \\) in \\( K1 \\), we have that \\( R(h(a_1), ..., h(a_n)) \\) is a fact in \\( K2 \\).

We can read fact as just a relation.

If we restrict the relation \\( R \\) in the definition of homomorphism above to a binary transitive, reflexive and anti-symmetric relation, we have a monotone map.

How are homomorphism and monotonicity related?

When I look up the definition of homomorphism on Wikipedia, it looks different from the one above:

$$ f ( x \star y ) = f(x) \star f(y) $$

How do I conciliate these definitions?