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O-MINIMAL STRUCTURES
R
An L-structure is o-minimal if every
definable subset of R is the union of finitely
many points and open intervals (a, b), where
a < b and a, b ∈ R ∪ {±∞}.
Examples (thus far)
• Rlin , the semilinear context
• Ralg , the semialgebraic context
Why “o-minimal”?
• o-minimal is short for ordered minimal.
• ordered minimal because the definable
subsets of R are exactly those that
must be there because of the presence
of <.
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Theme The hypothesis of o-minimality
combined with the power of definability
have remarkable consequences.
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Monotonicity Theorem. Let be an Lstructure that is o-minimal. Suppose that
f : R → R is -definable. Then there are
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−∞ = a0 < a1 < · · · < ak−1 < ak = ∞
in R ∪ {±∞} such that for each j < k either
f (aj , aj+1 ) is constant or is a strictly
monotone bijection of (possibly unbounded)
open intervals in R.
• In particular, all definable f : R →
are piecewise continuous.
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CELLS
in
R
in
R2
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R
More formally, let be an L-structure.
The collection of -cells is a subcollection
C = ∪∞
-definable subsets of
n=1 Cn of the
Rn for n = 1, 2, 3, ... defined recursively as
follows.
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R
Cells in R
The collection of cells C1 in R consists
of all single point sets {r} ⊂ R and all
open intervals (a, b) ⊆ R, where a < b
and a, b ∈ R ∪ {±∞}.
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Cells in
Rn+1
Assume the collection of cells Cn in Rn
have been defined. The collection Cn+1
of cells in Rn+1 consist of two different
kinds:
Graphs
Let C ∈ Cn and let f : C ⊆ R →
R be -definable and continuous.
Then Graph(f ) ⊆ Rn+1 is a cell;
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Generalized Cylinders
Let C ∈ Cn . Let f, g: C ⊆ R →
R be -definable and continuous
such that f (x̄) < g(x̄) for all x̄ ∈
C. Then the cylinder set (f, g)C ⊂
Rn+1 is a cell.
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Some Elementary Properties of Cells
• Cells are
R-definable.
• Cells are connected.
• Dimension for cells.
For each cell C ⊆ Rn there is a largest
k ≤ n and i1 , . . . , ik ∈ {1, 2, . . . , n}
such that if π: Rn → Rk is the
projection mapping given by
π(x1 , . . . , xn ) = (xi1 , . . . , xik ),
then π(C) ⊆ Rk is an open cell in Rk .
This value of k we call the dimension
of C.
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Cell Decomposition
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An ( -)decomposition D of Rn is a
partition of Rn into finitely many -cells
satisfying:
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If n = 1, then D consists of finitely
many open intervals and points;
If n > 1 and πn : Rn → Rn−1 denotes projection onto the first n − 1
coordinates, then {πn (C) : C ∈ D} is a
decomposition of Rn−1 .
Cell Decomposition Theorem. Let
be o-minimal and let S ⊂ Rn be
definable. Then there is a decomposition
D of Rn every definable set S ⊂ M n can
be partitioned definably into finitely many
cells. In particular, if f : A ⊂ Rn → R is
definable, then there is a partition of A into
cells such that the restriction of f to each
cell is continuous.
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Some Obvious Consequences
• Using the dimension of a cell as
defined above, we obtain a good geometric definition of the dimension of a
definable set.
• Since cells are connected, it follows that
every definable set has finitely many
connected components.
• The topological closure of a definable
set consists of finitely many connected
components; same for the interior.
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Definable families
Let S ⊂ Rn+p be a definable set in the
o-minimal structure . For each b̄ ∈ Rn
define
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Sb̄ := {ȳ ∈ Rp | (b̄, ȳ) ∈ S}.
Note Some Sb̄ may be empty.
The family {Sb̄ | b̄ ∈ Rn } of subsets of
called a definable family.
Rp is
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Uniform Bounds Theorem. Let
be
o-minimal and let S ⊂ Rn+1 be a definable
set such that Sb̄ is finite for all b̄ ∈ Rn . Then
there is a fixed K ∈ N satisfying
|Sb̄ | ≤ K for all b̄ ∈ M n .
Note A definable subset of R in an ominimal structure
is infinite if and only
if it contains an interval. So the theorem
actually is stronger.
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Quantifier elimination
is not easy to come by
Theorem (van den Dries). Let I be
an index set and for each i ∈ I let
fi : Rni → R be (total) analytic functions.
Then the structure (Ralg , {fi : i ∈ I} admits
quantifier elimination if and only if each fi
is semialgebraic.
So, e.g., Rexp , the real exponential field does
not have quantifier elimination.
Partial Elimination
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Suppose that a structure has the property
that every definable set is definable by
an existential formula, that is, a formula
having the form
∃x1 ∃x2 · · · ∃xk ϕ
where ϕ is a quantifier-free L-formula.
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How can this help?
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• Suppose that the
-definable sets
that are definable using quantifier-free
formulas can be analyzed, and that
all such have finitely many connected
components.
• The continuous image of a connected
set is connected (elementary topology).
• Existential quantification corresponds
to projection, and projection is a
continuous map.
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• Thus all -definable subsets of R have
finitely many connected components,
that is, all such are the union of finitely
many points and open intervals.
• Conclusion
is o-minimal, and
so all the geometric and topological
properties available as consequences of
o-minimality apply.
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Some References
L. van den Dries, Tame Topology and Ominimal Structures (London Mathematical
Society Lecture Note Series, vol. 248),
Cambridge: Cambridge University Press,
1998.
J. Knight, A. Pillay, and C. Steinhorn,
Definable sets in ordered structures II,
Transactions of the A. M. S., 295 (1986)
593–605.
A. Pillay and C. Steinhorn, Definable
sets in ordered structures I, Transactions of
the A. M. S., 295 (1986) 565–592.