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Almost o-minimal structures and
X
\mathfrak X
X
-structures
Authors
Masato Fujita
Publication date
3 April 2021
Publisher
View
on
arXiv
Abstract
We propose new structures called almost o-minimal structures and
X
\mathfrak X
X
-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's
X
\mathfrak X
X
-sets and
Y
\mathfrak Y
Y
-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an
X
\mathfrak X
X
-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded
X
\mathfrak X
X
-definable sets are definable in the structure. Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups
M
=
(
M
,
<
,
0
,
+
,
β¦
)
\mathcal M=(M,<,0,+,\ldots)
M
=
(
M
,
<
,
0
,
+
,
β¦
)
. Let
{
A
Ξ»
}
Ξ»
β
Ξ
\{A_\lambda\}_{\lambda\in\Lambda}
{
A
Ξ»
β
}
Ξ»
β
Ξ
β
be a finite family of definable subsets of
M
m
+
n
M^{m+n}
M
m
+
n
. Take an arbitrary positive element
R
β
M
R \in M
R
β
M
and set
B
=
]
β
R
,
R
[
n
B=]-R,R[^n
B
=
]
β
R
,
R
[
n
. Then, there exists a finite partition into definable sets \begin{equation*} M^m \times B = X_1 \cup \ldots \cup X_k \end{equation*} such that
B
=
(
X
1
)
b
βͺ
β¦
βͺ
(
X
k
)
b
B=(X_1)_b \cup \ldots \cup (X_k)_b
B
=
(
X
1
β
)
b
β
βͺ
β¦
βͺ
(
X
k
β
)
b
β
is a definable cell decomposition of
B
B
B
for any
b
β
M
m
b \in M^m
b
β
M
m
and either
X
i
β©
A
Ξ»
=
β
X_i \cap A_\lambda = \emptyset
X
i
β
β©
A
Ξ»
β
=
β
or
X
i
β
A
Ξ»
X_i \subseteq A_\lambda
X
i
β
β
A
Ξ»
β
for any
1
β€
i
β€
k
1 \leq i \leq k
1
β€
i
β€
k
and
Ξ»
β
Ξ
\lambda \in \Lambda
Ξ»
β
Ξ
. Here, the notation
S
b
S_b
S
b
β
denotes the fiber of a definable subset
S
S
S
of
M
m
+
n
M^{m+n}
M
m
+
n
at
b
β
M
m
b \in M^m
b
β
M
m
. We introduce the notion of multi-cells and demonstrate that any definable set is a finite union of multi-cells in the course of the proof of the above theorem.Comment: arXiv admin note: text overlap with arXiv:1912.0578
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oai:arXiv.org:2104.01312
Last time updated on 08/04/2021