# Definitions of isomorphism and elementary substructures

Let us define -all the definitions are from V. Manca, Logica matematica, 2001, 'mathematical logic'- a $\Sigma$-morphism as a function $f:D_1\to D_2$ between models $\mathscr{M}_1$ and $\mathscr{M}_2$, having respective universes $D_1$ and $D_2$, which preserves all the atomic formulas of model $\mathscr{M}_1$. Here I am not sure of a thing: atomic formulas can contain free variables, but can $f$ preserve a formula where free variables appears? If it can, I am not sure I understand what preservation is.

The definition of a $\Sigma$-isomorphism that I have in my book is that of a bijective $\Sigma$-morphism.

Let us define an elementary $\Sigma$-embedding as an injective $\Sigma$-morphism between models $\mathscr{M}_1$ and $\mathscr{M}_2$ preserving all the formulas of model $\mathscr{M}_1$.

Let us define an elementary $\Sigma$-substructure $\mathscr{M}$ of $\mathscr{M}'$ as a $\Sigma$-model $\mathscr{M}$ whose universe $D$ is a subset of the universe $D'$ of $\mathscr{M}'$ and whose inclusion function $\iota: D\hookrightarrow D'$ is an elementary $\Sigma$-embedding.

I read, in the same book, that $\mathscr{M}$ is elementarily $\Sigma$-embedded into $\mathscr{M}'$ if and only if $\mathscr{M}$ is $\Sigma$-isomorphic to an elementary $\Sigma$-substructure of $\mathscr{M}'$.

I have found alternative definitions of isomorphism where it is implied that an isomorphism preserves all the formulas of the model. Is this definition equivalent to requiring that it preserves the atomic formulas?

if it is not, I understand that $\mathscr{M}$ is elementarily $\Sigma$-embedded into $\mathscr{M}'$ only if $\mathscr{M}$ is $\Sigma$-isomorphic to an elementary $\Sigma$-substructure of $\mathscr{M}'$, but I do not understand why $\mathscr{M}$ is elementarily $\Sigma$-embedded into $\mathscr{M}'$ if $\mathscr{M}$ is $\Sigma$-isomorphic to an elementary $\Sigma$-substructure of $\mathscr{M}'$; how could it be proved? I thank you very much for any answer!

• About the meaning of "preservation". The truth of a formula $\phi(x)$ is preserved by $f:D_1\to D_2$ if $M_1\models\phi(a)\Rightarrow M_2\models\phi(fa)$ for every tuple $a\in D^{|x||}$. – Primo Petri Feb 12 '15 at 19:37
• An equivalent way of defining elementary embedding is as follows. We say that $f:D_1\to D_2$ is an elementary embedding if it is total and preserves the truth of all formulas. When $D_1\subseteq D_2$ and $f$ is the identity map, we say that $M_1\preceq M_2$. (You may find this formulation more clear. It is equivalent to what you write.) – Primo Petri Feb 12 '15 at 19:41