I'm reading about homomorphism preservation theorems and it starts with the following definitions that hold true in non-finite model theory:
Theorem 1.1 (Łoś-Tarski Theorem). A first-order formula is preserved under embeddings on all structures if, and only if, it is logically equivalent to an existential formula.
Theorem 1.2 (Lyndon's Positivity Theorem). A first-order formula is preserved under surjective homomorphisms on all structures if, and only if, it is logically equivalent to a positive formula.
Theorem 1.3 (Homomorphism Preservation Theorem). A first-order formula is preserved under homomorphisms on all structures if, and only if, it is logically equivalent to an existential-positive formula.
I've gone and found proofs for Theorems 1.1 and 1.2, but could not find a reference for Theorem 1.3. I can only assume that 1.3 is a direct consequence of combining 1.1 and 1.2 (maybe it seemed too trivial to cite?)
At first glance, 1.1 and 1.2 together say that surjective homomorphisms and embeddings both preserve existential-positive formulas; their converses together say that the homomorphisms preserving existential-positive formulas are the surjective embeddings (isomorphisms). I'm confused -- how does that lead us to 1.3's if-and-only-if claim of preservation under arbitrary homomorphisms?
The only way I can think of proving this in a way that's too trivial to cite is if every homomorphism in model theory can decompose into an embedding and a surjective homomorphism. Because if every homomorphism was decomposable into an embedding and a surjective part (like in group theory/ring theory/etc), that would make 1.3 obvious: an arbitrary homomorphism preserves existential-positive formulas since its surjective and embedding parts both preserve existential-positive formulas. Is this true in model theory? Can every homomorphism can be described by embeddings and surjective homomorphisms?
Yes, exactly as in the case of groups: if $h:\mathfrak{A}\rightarrow\mathfrak{B}$ is a homomorphism, then its image $\mathfrak{C}:=im(h)$ is a substructure of $\mathfrak{B}$, and we can think of $h$ as the composition of a surjection $\mathfrak{A}\rightarrow\mathfrak{C}$ (the "corestriction" of $h$ itself) and an embedding $\mathfrak{C}\rightarrow\mathfrak{B}$ (the inclusion).