# model

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

$$\mathcal{M}\vDash \phi ,$$ |

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

More generally, we say that a $\tau $-structure $\mathcal{M}$ is a *model* of a theory $T$ over $\tau $, if $\mathcal{M}\vDash \phi $ for every $\phi \in T$. 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

$$\mathcal{M}\vDash T.$$ |

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 |