PlanetMath: model

(more info)

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model

(Definition)

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

$\displaystyle \mathcal{M}\models \varphi,$

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 over $\tau$ , if $\mathcal{M}\models \varphi$ for every $\varphi\in T$ . When $\mathcal{M}$ is a model of , we say that satisfies $\mathcal{M}$ , or that $\mathcal{M}$ is satisfied by , and is written

$\displaystyle \mathcal{M}\models T.$

Example. Let $\tau=\lbrace \cdot \rbrace$ , where $\cdot$ is a binary operation symbol. Let be variables and

$\displaystyle T=\lbrace \forall x \forall y \forall z \left((x\cdot y)\cdot z=x\cdot (y\cdot z)\right) \rbrace.$

Then it is easy to see that any model of

is a semigroup, and vice versa.

Next, let $\tau'=\tau\cup \lbrace e\rbrace$ , where is a constant symbol, and

$\displaystyle T'=T\cup \lbrace \forall x (x\cdot e=x), \forall x\exists y (x\cdot y=e)\rbrace.$

Then

is a model of

iff

is a group. Clearly any group is a model of

. To see the converse, let

be a model of

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

for the product $x\cdot y$ . For any $x\in G$ , let $y\in G$ such that

and $z\in G$ such that

. Then

, so that $1x=1(1z)=(1\cdot 1)z=1z=x$ . This shows that

is the identity of

with respect to $\cdot$ . In particular,

, which implies

, or that

is a inverse of

with respect to $\cdot$ .

Remark. Let be a theory. A class of $\tau$ -structures is said to be axiomatized by if it is the class of all models of . is said to be the set of axioms for this class. This class is necessarily unique, and is denoted by $\operatorname{Mod}(T)$ . When consists of a single sentence $\varphi$ , we write $\operatorname{Mod}(\varphi)$ .