Definition:Theory (Logic)/Structure

Definition
Let $\LL$ be a logical language. Let $\MM$ be an $\LL$-structure.

The $\LL$-theory of $\MM$ is the $\LL$-theory consisting of those $\LL$-sentences $\phi$ such that:
 * $\MM \models \phi$

where $\models$ denotes that $\MM$ is a model for $\phi$.

This theory can be denoted $\map {\operatorname{Th} } \MM$ when the language $\LL$ is understood.