FANDOM


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. $