Łoś–Tarski preservation theorem

Remarks

The direction from 2 to 1 holds because universal sentences pass from a structure to its substructures. For the other direction, let $U$ be the set of universal sentences that follow from $T$, and let $A \models U$. Expand the signature by a constant for each element of $A$, and let $D$ be the set of quantifier-free sentences true in $A$, its diagram. If $T \cup D$ had no model, then by the compactness theorem $T$ would prove the negation of a finite conjunction from $D$. Replacing the new constants by variables gives a universal sentence in $U$ that fails in $A$. So $T \cup D$ has a model $B$, into which $A$ embeds, and condition 1 gives $A \models T$.

A version for formulas holds too: a formula is preserved by substructures between models of $T$ exactly when it is equivalent modulo $T$ to a universal formula. Dually, the class of models of $T$ is closed under extensions exactly when $T$ is equivalent to a set of existential sentences.

Example: the theory of groups in the signature $(\cdot, {}^{-1}, e)$ is closed under substructures, and the group axioms are indeed universal. In the signature $(\cdot)$ alone, a substructure of a group is only a subsemigroup, such as $\N_{>0}$ in $(\Z, +)$, and the axioms of groups in this signature need existential quantifiers.

The theorem uses compactness, and it fails when only finite structures are considered.

Going deeper

For a student

The Łoś–Tarski theorem is the first of the preservation theorems of model theory. Others in the same spirit are the Chang–Łoś–Suszko theorem, for unions of chains and $\forall\exists$ sentences, and Lyndon's theorem, for homomorphic images and positive sentences. In universal algebra it explains why an elementary class of algebras closed under subalgebras has a universal axiomatization. This is a first step towards the descriptions of quasivarieties and varieties. The proof method, realizing the diagram of a structure in a model of $T$, is the method of diagrams used throughout model theory.