Simple closure ($M{\preccurlyeq }_{s}N$)

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.