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⊆S we have either C is finite or S∖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≡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 |
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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 |