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 $\mathrm{Mod}(\phi )$.

Title	model
Canonical name	Model
Date of creation	2013-03-22 13:00:14
Last modified on	2013-03-22 13:00:14
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	33
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03C95
Related topic	Structure
Related topic	SatisfactionRelation
Related topic	AlgebraicSystem
Related topic	RelationalSystem
Defines	model