Note, Daniel, that my definition of "homomorphism" makes its resemblance to a monotone map a bit clearer. Using my notation, Wikipedia told you that a homomorphism \\(f : A \to B\\) obeys

$$ f(a \ast_A a') = f(a) \ast_B f(a') $$

for all \\(a,a' \in A\\). This is equivalent to what I said:

$$ a \ast_A a' = a'' \textrm{ implies } f(a) \ast_B f(a') = f(a'') $$

for all \\(a,a',a'' \in A\\). The Wikipedia definition is more efficient and more commonly used, but my definition shows what's going on: we're taking a fact that holds in \\(A\\), apply \\(f\\) to each of the elements involved, and claim that the corresponding fact must hold in \\(B\\). That's the general idea of a "structure-preserving map", of which homomorphisms and monotone maps are special cases.

This becomes an incredibly important theme in category theory, and one could write a book about it. My mistake in Problem 11 shows one of the subtleties: operations work a bit differently than relations.

$$ f(a \ast_A a') = f(a) \ast_B f(a') $$

for all \\(a,a' \in A\\). This is equivalent to what I said:

$$ a \ast_A a' = a'' \textrm{ implies } f(a) \ast_B f(a') = f(a'') $$

for all \\(a,a',a'' \in A\\). The Wikipedia definition is more efficient and more commonly used, but my definition shows what's going on: we're taking a fact that holds in \\(A\\), apply \\(f\\) to each of the elements involved, and claim that the corresponding fact must hold in \\(B\\). That's the general idea of a "structure-preserving map", of which homomorphisms and monotone maps are special cases.

This becomes an incredibly important theme in category theory, and one could write a book about it. My mistake in Problem 11 shows one of the subtleties: operations work a bit differently than relations.