strongly minimal

Let L be a first order language and let M be an L-structureMathworldPlanetmath. Let S, a subset of the domain of M be a definable infinite setMathworldPlanetmath. Then S is minimalPlanetmathPlanetmath iff every definable CS we have either C is finite or SC is finite. We say that M is minimal iff the domain of M is a strongly minimal set.

We say that M is strongly minimal iff for every NM, we have that N is minimal. Thus if T is a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath L theory then we say T is strongly minimal if it has some model (equivalently all models) which is strongly minimal.

Note that M is strongly minimal iff every definable subset of M is quantifier free definable in a languagePlanetmathPlanetmath with just equality. Compare this to the notion of o-minimal structures.

Title strongly minimal
Canonical name StronglyMinimal
Date of creation 2013-03-22 13:27:13
Last modified on 2013-03-22 13:27:13
Owner Timmy (1414)
Last modified by Timmy (1414)
Numerical id 5
Author Timmy (1414)
Entry type Definition
Classification msc 03C07
Classification msc 03C10
Classification msc 03C45
Related topic OMinimality
Defines strongly minimal
Defines minimal