Substructure of a first-order structure ($A\subseteq B$)

An $L$-substructure of a first-order structure $A\subseteq B$ has a nonempty domain containing all named constants and closed under the functions of $B$, and each relation of $A$ is the restriction of the corresponding relation of $B$. Consequently every quantifier-free formula with parameters in $A$ has the same truth value in $A$ and $B$. This is weaker than being an elementary substructure, which preserves all first-order formulas.

The values of all closed terms in a first-order structure form a substructure when the first-order language has at least one constant. The constant ensures nonemptiness, and applying a function symbol to closed-term values again gives a closed-term value. Relations are restricted from the ambient first-order structure. Every universal sentence true in the ambient first-order structure remains true in this substructure.

A substructure $M\subseteq N$ is simply closed in $N$ when every formulawith quantifier-free formula $\theta$ and parameters $\mathbf{a}$ from $M$, that is true in $N$ is already true in $M$. This is existential closure in the specified extension. It is weaker than being an elementary substructure, which preserves arbitrary first-order formulas.

One quantified variable with a quantifier-free formula as matrix does not suffice. Let $M$ be an infinite edgeless simple graph, and let $N$ add two new vertices joined to each other and to no old vertex. Every one-variable quantifier-free formula over a finite parameter tuple from $M$ realized by a new vertex is realized by an old vertex outside that tuple. Nevertheless $N$ satisfies $\exists x\exists yE(x,y)$ and $M$ does not. Iteration of one-variable reflection fails because the remaining existential matrix need not be a quantifier-free formula.

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.