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单词 Type
释义

type


Let L be a first order language.Let M be an http://planetmath.org/node/3384L-structureMathworldPlanetmath.Let BM, and let aMn.Then we define the type of a over B to be the set of L-formulasMathworldPlanetmath ϕ(x,b¯) with parameters b¯ from B so that Mϕ(a,b¯).A collectionMathworldPlanetmath of L-formulas is a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath n-type over B iff it is of the above form for some B,M and aMn.

We call any consistent collection of formulas p in n variables with parameters from B a partial n-type over B. (See criterion for consistency of sets of formulas.)

Note that a complete n-type p over B is consistent so is in particular a partial type over B. Also p is maximal in the sense that for every formula ψ(x,b¯) over B we have either ψ(x,b¯)p or ¬ψ(x,b¯)p.In fact, for every collection of formulas p in n variables the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  • p is the type of some sequence of n elements a over B in some model NM

  • p is a maximal consistent set of formulas.

For nω we define Sn(B) to be the set of complete n-types over B.

Some authors define a collection of formulas p to be a n-type iff p is a partial n-type. Others define p to be a type iff p is a complete n-type.

A type (resp. partial type/complete type) is any n-type (resp. partial type/complete type) for some nω.

Titletype
Canonical nameType
Date of creation2013-03-22 13:22:45
Last modified on2013-03-22 13:22:45
Ownerratboy (4018)
Last modified byratboy (4018)
Numerical id6
Authorratboy (4018)
Entry typeDefinition
Classificationmsc 03C07
Related topicFormula
Related topicDefinableType
Related topicTermsAndFormulas
Definestype
Definescomplete type
Definespartial type
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