Let τ be a signaturePlanetmathPlanetmathPlanetmath and φ be a sentenceMathworldPlanetmath over τ. A structureMathworldPlanetmath ( for τ is called a model of φ if


where is the satisfaction relation. When φ, we says that φ satisfies , or that is satisfied by φ.

More generally, we say that a τ-structure is a model of a theory T over τ, if φ for every φT. When is a model of T, we say that T satisfies , or that is satisfied by T, and is written


Example. Let τ={}, where is a binary operationMathworldPlanetmath symbol. Let x,y,z be variables and


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

Next, let τ=τ{e}, where e is a constant symbol, and


Then G is a model of T iff G is a group. Clearly any group is a model of T. To see the converseMathworldPlanetmath, let G be a model of T and let 1G be the interpretationMathworldPlanetmath of eτ and :G×GG be the interpretation of τ. Let us write xy for the productMathworldPlanetmathPlanetmath xy. For any xG, let yG such that xy=1 and zG such that yz=1. Then 1z=(xy)z=x(yz)=x1=x, so that 1x=1(1z)=(11)z=1z=x. This shows that 1 is the identityPlanetmathPlanetmathPlanetmath of G with respect to . In particular, x=1z=z, which implies 1=yz=yx, or that y is a inversePlanetmathPlanetmath of x with respect to .

Remark. Let T be a theory. A class of τ-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 Mod(T). When T consists of a single sentence φ, we write Mod(φ).

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