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For a first-order theory $T$ and $A\models T_{\forall }$, an algebraically prime extension is a structure embedding $i:A\to P\models T$ such that every embedding $j:A\to N\models T$ factors as $j=h\circ i$ for some structure embedding $h:P\to N$. The embeddings need not be elementary or unique. A theory has algebraically prime models when every such $A$ has an extension of this kind.

An embedding of first-order structures is existential when every existential formula with parameters from the source has the same truth value in the source and target. Forward preservation follows from being a structure embedding; reflection is the additional requirement. Applying compactness theorem to the target's diagram of a structure and the source's elementary diagram places the target inside an elementary extension of the source.

An existential formula is an existential block followed by a quantifier-free formula, with remaining free variables serving as parameters. Truth is preserved by structure embeddings. Reflection of truth for every such formula defines an existential embedding.

A finite conjunction from the signed atomic diagram of $A$ becomes, after existentially closing its new names, a sentence in its existential theory. Joint satisfiability with $T$ therefore makes the full diagram finitely satisfiable with $T$. Compactness gives the embedding, with inequalities ensuring injectivity and signed atomic formulas ensuring both preservation and reflection of relations as required for a structure embedding. The reverse implication is preservation of existential truth under embeddings.

The existential theory of a structure is the set of existential sentences with quantifier-free matrices true in it. Its witnesses persist in extensions because quantifier-free truth is preserved by structure embeddings. It differs from an existential diagram, which additionally uses names for Elements.

Name every element $a\in A$ by a new constant $c_a$, distinct from the symbols already in $L^{\prime }$. Let $\mathrm{Diag}(A)$ be its full atomic and negated atomic diagram of a structure. A model of this diagram contains an isomorphic copy of $A$ as an $L$-substructure. We claimis satisfiable.

Otherwise compactness theorem gives a finite diagram conjunction $\delta (c_{a_1},\dots ,c_{a_m})$ inconsistent with $T^{\prime }$. Replace these new constants by variables. Since $T^{\prime }$ says nothing about the new names, it then entails the universal sentenceThe conjunction $\delta$ uses only symbols of $L$, so this is one of the universal $L$-consequences assumed true in $A$. But $A$ satisfies $\delta (a_1,\dots ,a_m)$, a contradiction. A model of the consistent enlarged theory gives an embedding of $A$ into the $L$-reduct of a model of $T^{\prime }$. Negated atomic facts and disequalities in the diagram ensure that this is a structure embedding, not merely a homomorphism.

Let $U$ be all universal sentences true in $C$. The theory $T\cup U$ is consistent. Indeed, if a finite conjunction $u$ of sentences from $U$ were inconsistent with $T$, then $T\models \neg u$. A finite conjunction of universal sentences is universal, so its negation is equivalent to an existential sentence. By the hypothesis $C$ would satisfy $\neg u$, whereas it satisfies $u$.

Choose $A\models T\cup U$. Now expand the language by names for every element of $C$ and consider its elementary diagram $\mathrm{ElDiag}(C)$. Its universal consequences in the original language are exactly $U$: any such consequence is true in the named copy of $C$, while any original-language sentence true in $C$ already belongs to its elementary diagram. Part (b), applied to $A$ and this expanded theory, gives a structure embedding of $A$ into a model $D$ of $\mathrm{ElDiag}(C)$. The named copy of $C$ is an elementary substructure of $D$. Identifying that copy with $C$, we obtainThus the required target is an elementary extension of $C$, although $C$ itself need not be a model of $T$.

A 1-embedding, or existential embedding, is a structure embedding $\alpha :A\to B$ satisfyingfor every quantifier-free $\theta$ and every parameter tuple $\mathbf{a}$ from $A$. The forward direction is automatic for an embedding; reflection of existential formulas is the extra property. For an inclusion this says that $A$ is simply closed in $B$. This is weaker than requiring an elementary embedding, which preserves all formulas.

Let $T_{\forall }$ denote the universal consequences of a theory: all universal first-order sentences entailed by $T$. A first-order structure satisfies $T_{\forall }$ exactly when it embeds into a model of $T$, by the compactness theorem applied to its diagram of a structure.

The theory has algebraically prime models if, for every $\mathcal{A}\models T_{\forall }$, there are $\mathcal{P}\models T$ and a structure embedding $i:\mathcal{A}\to \mathcal{P}$ such that every structure embedding $j:\mathcal{A}\to \mathcal{N}$, $\mathcal{N}\models T$, factors as $j=h\circ i$ for some structure embedding $h:\mathcal{P}\to \mathcal{N}$. Neither $i$ nor $h$ is required to be elementary.

For $\mathcal{M}\subseteq \mathcal{N}$, simple closure means that every existential quantifier-free formula over $M$ which has a witness in $N$ has one in $M$:Now take two models $\mathcal{M},\mathcal{N}\models T$ with common substructure $\mathcal{A}$. Since $A$ embeds into $M$, it satisfies $T_{\forall }$. Choose its algebraically prime extension $\mathcal{P}$, and embed $P$ into both $M$ and $N$ over $A$.

If $\exists x\theta (x,\bar{a})$ holds in $M$, the image of $P$ in $M$ is a model of $T$. The assumed simple closure of this image transfers a witness from $M$ into $P$. Its embedding into $N$ then transfers the quantifier-free formula and its witness into $N$. Thus the hypothesis of QET1 is satisfied. The second test follows: $T$ has quantifier elimination.

A torsion-free divisible Abelian group is naturally a vector space over the rational numbers: for $m\in \mathbb{Z}$ and $n>0$, define $(m/n)a$ as the unique $b$ with $nb=ma$. Divisibility supplies existence and torsion-freeness supplies uniqueness.

In the usual group first-order language $\{+,-,0\}$, a model $A$ of the universal part of DAG is a torsion-free group which is Abelian. If the language instead uses only $\{+,0\}$, a substructure can be merely a torsion-free cancellative commutative monoid. Handle this convention by first taking its Grothendieck group $G(A)$: its elements are formal differences $a-b$, withThe cancellative commutative monoid condition makes $A\to G(A)$ injective. If $n(a-b)=0$, then $na=nb$, and torsion-freeness gives $a=b$; hence $G(A)$ is a torsion-free abelian group. In the full group language simply take $G(A)=A$.

For $A\ne \{0\}$ form its rational divisible hullConcretely its elements are fractions $g/n$ with $n>0$, where $g/n=h/m$ if $mg=nh$. The canonical embedding of $A$ into $P$ is injective, and $P$ is nontrivial, divisible, Abelian and torsion-free, so it satisfies DAG.

Let $j:A\to N$ be any structure embedding into a model of DAG. Extend it first to formal differences if necessary. Its unique extension to the rational divisible hull sends $g/n$ to the unique element $b\in N$ with $nb=j(g)$. This is a group homomorphism fixing the given copy of $A$. It is injective: an element mapped to zero has $j(g)=0$, hence $g=0$. Thus every embedding into a DAG model factors through $P$.

The zero case must be treated separately: its rational divisible hull is zero and does not satisfy DAG. Instead choose $P=(\mathbb{Q},+)$. Given any nontrivial DAG model $N$, choose $b\ne 0$; the map $q\mapsto qb$ embeds $P$ into $N$ over zero. Therefore DAG has algebraically prime models, including over the trivial base.

A first-order theory $T$ is model-complete if every structure embedding between models of $T$ is an elementary embedding. Equivalently, whenever $\mathcal{M}\subseteq \mathcal{N}$ and both satisfy $T$, one has $\mathcal{M}\preccurlyeq \mathcal{N}$:for every first-order formula $\phi$ and every finite tuple $\bar{a}\in M$. This is preservation of all formulas with parameters, rather than merely all first-order sentences.

Assume $T$ has quantifier elimination, and let $j:\mathcal{M}\to \mathcal{N}$ be a structure embedding between its models. For any first-order formula $\phi (\bar{x})$, choose a quantifier-free formula $\psi$ equivalent to it modulo $T$. A structure embedding preserves and reflects atomic formulas; induction through the Boolean connectives therefore preserves every quantifier-free formula. ThusThe embedding is elementary. Every theory with quantifier elimination is model-complete.

An $L$-embedding $\pi :A\to B$ is an injective map preserving the interpretations of all constants and functions and preserving and reflecting every relation. Explicitly $\pi (c^A)=c^B$, $\pi (f^A(\bar{a}))=f^B(\pi \bar{a})$, and $A\models R(\bar{a})$ if and only if $B\models R(\pi \bar{a})$. Injectivity handles logical equality. Equivalently it is an isomorphism onto an $L$-substructure of $B$, or a map preserving the truth of all quantifier-free formulas. Preserving relations in only one direction would instead give a weaker homomorphism notion, not an embedding.

If a model $B$ of $T$ contains $A$ as a substructure, every existential sentence true in $A$ remains true in $B$: use the same witness tuple and preservation of quantifier-free truth. Hence $B$ satisfies the required joint theory.

For the existential-theory embedding criterion, conversely suppose that joint theory has a model. Expand the language by a constant $c_a$ for each $a\in A$, and add the atomic and negated atomic diagram $D(A)$ to $T$. Any finite collection of its literals is a quantifier-free conjunction $\theta (\bar{c})$ true of the corresponding tuple in $A$. Replacing those named constants by variables and existentially closing gives a sentence in ${\mathrm{Th}}_{\exists }(A)$. A model of the assumed joint theory can supply its witnesses, and hence an expansion satisfying those finitely many diagram requirements and any finite subset of $T$.

Compactness therefore gives a model of $T+D(A)$. Its named Elements define an embedding of $A$ into its $L$-reduct: inequalities ensure injectivity, and all function and relation literals ensure preservation and reflection. Replace the ambient model by an isomorphic copy identifying this image with $A$ itself. Thus a model of $T$ containing $A$ exists. The diagram step explains how parameter-free existential information nevertheless yields an embedding of the whole structure.

If $B\models T$ and $B\subseteq A^{\ast }\equiv A$, every universal sentence of $A$ holds in $A^{\ast }$ and is preserved by passage to the substructure $B$. Thus $B$ witnesses satisfiability of the joint theory.

For universal theory and substructures of elementarily equivalent models, conversely let $B$ satisfy $T$ together with the universal theory of $A$. Every existential sentence true in $B$ must be true in $A$: if an existential $\sigma$ were false in $A$, its universal negation would belong to ${\mathrm{Th}}_{\forall }(A)$ and hold in $B$, a contradiction. Hence ${\mathrm{Th}}_{\exists }(B)\subseteq \mathrm{Th}(A)$. In particular $\mathrm{Th}(A)\cup {\mathrm{Th}}_{\exists }(B)$ is satisfiable, with model $A$. Apply the preceding embedding result to the theory $\mathrm{Th}(A)$ and structure $B$. It supplies a structure $A^{\ast }\models \mathrm{Th}(A)$ containing $B$. Therefore $A^{\ast }\equiv A$ and $B\subseteq A^{\ast }$, as required. Elementary equivalence here does not assert that $A$ itself contains a copy of $B$.

Existential sentences are preserved under extensions: witnesses remain present and quantifier-free truth is unchanged by an embedding. Therefore any existential axiomatization implies the specified extension-closure property.

For existential axiomatization from extension closure, let $E$ be all existential consequences of $T$, and suppose $A\models E$. We claim that $T\cup {\mathrm{Th}}_{\forall }(A)$ is satisfiable. Otherwise compactness supplies finitely many universal sentences ${\theta }_1,\dots ,{\theta }_m$ true in $A$ such that $T\models \neg ({\theta }_1\wedge \cdots \wedge {\theta }_m)$. The finite conjunction is universal after renaming bound variables, so its negation is existential. It belongs to $E$ but is false in $A$, a contradiction.

The preceding universal-theory embedding result now yields $B\models T$ and $A^{\ast }\equiv A$ with $B\subseteq A^{\ast }$. By the assumed extension closure, $A^{\ast }\models T$; elementary equivalence implies $A\models T$. Thus $E$ axiomatizes exactly the models of $T$, proving existential axiomatizability is equivalent to closure under extensions. This includes the inconsistent case, whose empty model class is axiomatized by an existential contradiction.

A model is prime if it has an elementary embedding into every model of $T$. A model is atomic if every finite tuple realizes an isolated complete type over the empty set.

If $A$ is prime, the downward Löwenheim-Skolem theorem gives a countable model $M$ of $T$, and an elementary embedding $A\to M$ makes $A$ countable. If a tuple of $A$ realized a nonisolated type, the omitting types theorem would give a model of $T$ omitting that type. The prime embedding into it would preserve the tuple's full type and realize the omitted type, a contradiction. Thus a prime model is countable and atomic.

Conversely enumerate a countable atomic model as $a_0,a_1,\dots$, and take any $B\models T$. Build finite partial elementary maps into $B$. Suppose the current tuple $\bar{a}$ maps to $\bar{b}$. Choose a formula $\theta (\bar{x},y)$ isolating the type of $(\bar{a},a_n)$. The assertion $\exists y\theta (\bar{a},y)$ holds in $A$, so its counterpart holds at $\bar{b}$ in $B$ by partial elementarity. Choose a witness $b_n$. Since $\theta$ isolates the entire extended tuple type, the extended map is elementary on its finite domain; its equality formulas ensure distinct Elements remain distinct. The union is a total elementary embedding $A\to B$. The empty initial tuple works because $T$ is complete. Hence

Yes: $Q^{+}$ is prime. In any model of its complete diagram, map each rational to the interpretation of its named constant. The diagram ensures this is an embedding, and quantifier elimination for dense linear orders without endpoints makes it elementary. Alternatively every finite tuple of $Q^{+}$ has its complete type isolated by the conjunction equating its coordinates to their named rational constants. Thus it is a countable atomic model, and the earlier prime-model equivalence applies. A prime model may omit many types; primeness is not saturation.

Expand the language by a constant $c_a$ for every element of the structure. Its diagram of a structure consists of all atomic and negated atomic sentences true in this expanded structure. Its elementary diagram, also called the complete diagram, consists of all first-order sentences true in the expansion.

For a model $M$ of the diagram, the map $a\mapsto c_a^M$ is injective because the diagram includes all required inequalities. It preserves functions and constants by the corresponding equations, and it preserves and reflects relations by the positive and negative atomic sentences. Thus it is a structure embedding onto a substructure of the reduct of $M$. Conversely, a structure containing an isomorphic copy of the original as a substructure can interpret the new constants at that copy and satisfy its diagram.

For the elementary diagram the same argument works for every formula, not just atomic ones. The map is therefore an elementary embedding. Conversely any elementary embedding gives an expansion satisfying the elementary diagram. These are characterizations up to the named copy; a model of either diagram may have additional unnamed elements.

A model-complete theory is a first-order theory for which every structure embedding between two of its first-order models is an elementary embedding. Thus, if $f:\mathcal{A}\hookrightarrow \mathcal{B}$ is such an embedding, then for every first-order formula $\phi (\mathbf{x})$ and tuple $\mathbf{a}$ from $A$,first-order model completeness does not assert that every two first-order models are elementarily equivalent: it constrains embeddings that exist between first-order models, whereas completeness constrains all first-order models.