Model

Remarks

Example: let $T$ be the theory of groups in the signature $\set{\cdot, e, {}^{-1}}$. The structure $(\Z, +, 0, -)$, with $\cdot$ read as $+$, $e$ as $0$ and ${}^{-1}$ as $-$, is a model of $T$. Indeed addition of integers is associative, $0$ is neutral and $-x$ is an inverse of $x$.

Non-example: the structure $(\N, +, 0, \id)$, with ${}^{-1}$ read as the identity map, is not a model of $T$. The sentence $\forall x \, (x \cdot x^{-1} = e)$ fails at $x = 1$, because $1 + 1 \ne 0$.

A theory is satisfiable when it has a model. The class of all models of $T$ is often written $\mathrm{Mod}(T)$.