Theory Interpretation (Satisfiability Modulo Theory)

Definition

Theory Interpretation (Satisfiability Modulo Theory)

Let $\mathcal{T}$ be a theory over a signature $\Sigma$. A $\Sigma$-interpretation $I$ is a $\mathcal{T}$-interpretation, or a model of $\mathcal{T}$, if every axiom of $\mathcal{T}$ is valid in $I$:

$$I\models \mathcal{T}⟺\forall A\in \mathcal{T}:I\models A.$$