model

Let $\tau$ be a signature and $\phi$ be a sentence over $\tau$. A structure (http://planetmath.org/Structure) $ℳ$ for $\tau$ is called a model of $\phi$ if

$$ℳ\vDash \phi ,$$

where $\vDash$ is the satisfaction relation. When $ℳ\vDash \phi$, we says that $\phi$ satisfies $ℳ$, or that $ℳ$ is satisfied by $\phi$.

More generally, we say that a $\tau$-structure $ℳ$ is a model of a theory $T$ over $\tau$, if $ℳ\vDash \phi$ for every $\phi \in T$. When $ℳ$ is a model of $T$, we say that $T$ satisfies $ℳ$, or that $ℳ$ is satisfied by $T$, and is written

$$ℳ\vDash T.$$

Example. Let $\tau =\{\cdot \}$, where $\cdot$ is a binary operation symbol. Let $x,y,z$ be variables and

$$T=\{\forall x\forall y\forall z((x\cdot y)\cdot z=x\cdot (y\cdot z))\}.$$

Then it is easy to see that any model of $T$ is a semigroup, and vice versa.

Next, let ${\tau }^{\prime }=\tau \cup \{e\}$, where $e$ is a constant symbol, and

$$T^{\prime }=T\cup \{\forall x(x\cdot e=x),\forall x\exists y(x\cdot y=e)\}.$$

Then $G$ is a model of $T^{\prime }$ iff $G$ is a group. Clearly any group is a model of $T^{\prime }$. To see the converse, let $G$ be a model of $T^{\prime }$ and let $1\in G$ be the interpretation of $e\in {\tau }^{\prime }$ and $\cdot :G\times G\to G$ be the interpretation of $\cdot \in {\tau }^{\prime }$. 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 $\mathrm{Mod}(T)$. When $T$ consists of a single sentence $\phi$, we write $\mathrm{Mod}(\phi )$.