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Let $\tau$ be a signature and $\varphi$ be a sentence over $\tau$ . A structure $\mathcal{M}$ for $\tau$ is called a model of $\varphi$ if $$\mathcal{M}\models \varphi,$$ where $\models$ is the satisfaction relation. When $\mathcal{M}\models \varphi$ , we says that $\varphi$ satisfies $\mathcal{M}$ , or that $\mathcal{M}$ is satisfied by $\varphi$ .
More generally, we say that a $\tau$ -structure $\mathcal{M}$ is a model of a theory $T$ over $\tau$ , if $\mathcal{M}\models \varphi$ for every $\varphi\in T$ . When $\mathcal{M}$ is a model of $T$ , we say that $T$ satisfies $\mathcal{M}$ , or that $\mathcal{M}$ is satisfied by $T$ , and is written $$\mathcal{M}\models T.$$
Example. Let $\tau=\lbrace \cdot \rbrace$ , where $\cdot$ is a binary operation symbol. Let $x,y,z$ be variables and $$T=\lbrace \forall x \forall y \forall z \left((x\cdot y)\cdot z=x\cdot (y\cdot z)\right) \rbrace.$$ Then it is easy to see that any model of $T$ is a semigroup, and vice versa.
Next, let $\tau'=\tau\cup \lbrace e\rbrace$ , where $e$ is a constant symbol, and $$T'=T\cup \lbrace \forall x (x\cdot e=x), \forall x\exists y (x\cdot y=e)\rbrace.$$ Then $G$ is a model of $T'$ iff $G$ is a group. Clearly any group is a model of $T'$ . To see the converse, let $G$ be a model of $T'$ and let $1\in G$ be the
interpretation of $e\in \tau'$ and $\cdot:G\times G\to G$ be the interpretation of $\cdot\in \tau'$ . Let us write $xy$ for the product $x\cdot y$ . For any $x\in G$ , let $y\in G$ such that $xy=1$ and $z\in G$ such that $yz=1$ . Then $1z=(xy)z=x(yz)=x1=x$ , so that $1x=1(1z)=(1\cdot 1)z=1z=x$ . This shows that $1$ is the identity of $G$ with respect to $\cdot$ . In particular, $x=1z=z$ , which implies $1=yz=yx$ , or that $y$ is a inverse of $x$ with respect to $\cdot$ .
Remark. Let $T$ be a theory. A class of $\tau$ -structures is said to be axiomatized by $T$ if it is the class of all models of $T$ . $T$ is said to be the set of axioms for this class. This class is necessarily unique, and is denoted by $\operatorname{Mod}(T)$ . When $T$ consists of a single sentence $\varphi$ , we write $\operatorname{Mod}(\varphi)$ .
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