## FANDOM

78 Pages

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 $T.$