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strongly minimal (Definition)

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.



"strongly minimal" is owned by Timmy.
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See Also: o-minimality

Also defines:  strongly minimal, minimal
Keywords:  minimal

Attachments:
example of strongly minimal (Example) by CWoo
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Cross-references: structures, o-minimal, equality, language, quantifier free, theory, complete, finite, iff, infinite set, definable, domain, subset, first order language
There are 78 references to this entry.

This is version 2 of strongly minimal, born on 2003-02-11, modified 2004-07-11.
Object id is 4019, canonical name is StronglyMinimal.
Accessed 5118 times total.

Classification:
AMS MSC: 03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)
 03C10 (Mathematical logic and foundations :: Model theory :: Quantifier elimination, model completeness and related topics)
 03C45 (Mathematical logic and foundations :: Model theory :: Classification theory, stability and related concepts)
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