A ${\mathcal {L}}$ -structure $M$ is existentially closed in a class ${\mathcal {K}}$ of ${\mathcal {L}}$ -structures, if for any ${\mathcal {L}}$ -structure $N\in {\mathcal {K}}$ with $M\subseteq N$ we have that $M$ is 1-elementary in $N$ , i.e., for every existential ${\mathcal {L}}$ -formula $\exists {\bar {y}}\phi ({\bar {x}};{\bar {y}})$ and every ${\bar {a}}\subseteq M$ we have that

$N\models \exists {\bar {y}}\phi ({\bar {a}};{\bar {y}})\leftrightarrow M\models \exists {\bar {y}}\phi ({\bar {a}};{\bar {y}}).$

Also for a ${\mathcal {L}}$ -theory $T$ we say that $M$ is an existentially closed model of $T$ , if $M$ is existentially closed in the class of models of $T.$ ::: ::: :::