Existentially closed

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 $$T.$$