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.

A substructure $M\subseteq N$ is simply closed when each existential quantifier-free formula over $M$ realized in $N$ is already realized in $M$. A single quantified variable suffices in the definition; iterating covers finite tuples. This property is also called existential closure in this specified extension. It is weaker than being an elementary substructure.

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.