example of definable type

Consider (𝐐,<) as a structureMathworldPlanetmath in a languagePlanetmathPlanetmath with one binary relationMathworldPlanetmath, which we interpret as the order. This is a universalPlanetmathPlanetmath, 0-categorical structure (see example of universal structure).

The theory of (𝐐,<) has quantifier eliminationMathworldPlanetmath, and so is o-minimal. Thus a type over the set 𝐐 is determined by the quantifier free formulas over 𝐐, which in turn are determined by the atomic formulas over 𝐐. An atomic formula in one variable over B is of the form x<b or x>b or x=b for some bB. Thus each 1-type over 𝐐 determines a Dedekind cut over 𝐐, and conversely a Dedekind cut determines a complete type over 𝐐. Let D(p):={a𝐐:x>ap}.

Thus there are two classes of type over 𝐐.

  1. 1.

    Ones where D(p) is of the form (-,a) or (-,a] for some a𝐐. It is clear that these are definable from the above discussion.

  2. 2.

    Ones where D(p) has no supremum in 𝐐. These are clearly not definable by o-minimality of 𝐐.

Title example of definable type
Canonical name ExampleOfDefinableType
Date of creation 2013-03-22 13:29:43
Last modified on 2013-03-22 13:29:43
Owner aplant (12431)
Last modified by aplant (12431)
Numerical id 5
Author aplant (12431)
Entry type Example
Classification msc 03C07
Related topic ExampleOfUniversalStructure
Related topic DedekindCuts