strongly minimal

Let $L$ be a first order language and let $M$ be an $L$-structure. Let $S$, a subset of the domain of $M$ be a definable infinite set. Then $S$ is minimal iff every definable $C\subseteq S$ we have either $C$ is finite or $S\setminus C$ 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 $N\equiv M$, we have that $N$ is minimal. Thus if $T$ is a complete $L$ 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 language 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