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An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field
[article]
2019
arXiv
pre-print
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The proof is complicated by the fact that a definable open set need not be a union of finitely many open cells. ...
U is a finite union of regular open R-definable sets. ...
arXiv:1910.13504v1
fatcat:xsejcsy2m5eu5h45koqmyhb5dq
Structures having o-minimal open core
2009
Transactions of the American Mathematical Society
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By the Cell Decomposition Theorem, every set definable in an o-minimal structure is a Boolean combination of open definable sets. ...
If T is an extension of the theory of densely ordered groups such that every open unary set definable in any model of T is a finite union of open intervals, then every model of T has o-minimal open core ...
definable unary set is a finite union of open intervals. ...
doi:10.1090/s0002-9947-09-04908-3
fatcat:gqydu6uuezgandlqtgmd7r4cuu
Almost o-minimal structures and 𝔛-structures
[article]
2021
arXiv
pre-print
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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. ...
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 ...
In particular, the definable set ]c, d[∩D N is not a finite union of points and open intervals. Here, the notation D N be the subset of N defined by the same formula as D. ...
arXiv:2104.01312v1
fatcat:puwzoagquffhpphd2dsf5c7dbm
Locally o-minimal structures and structures with locally o-minimal open core
2013
Annals of Pure and Applied Logic
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We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core. ...
We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core. ...
Thus, by Lemma 2.11, it is clear that X is the union of a pseudofinite set and an open cell, and therefore X is the union of 2 multi-cells. ...
doi:10.1016/j.apal.2012.10.002
fatcat:fmwztdzvvva35prhg3qunus77i
A theorem about topological $n$-cells
1954
Proceedings of the American Mathematical Society
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as the union of a finite number of members of 2; (3) SiEPi for each *'. ...
It is easy to see that the closure of Sm -Pm~i can be expressed as the union of a finite set <ri, ■ ■ ■ , ap of elements of S. ...
doi:10.1090/s0002-9939-1954-0062434-9
fatcat:opns6p56xba3jeb5b7qzlgmgii
Extension of Lipschitz maps definable in Hensel minimal structures
[article]
2022
arXiv
pre-print
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It may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's theorem. ...
In this paper, we establish a theorem on extension of Lipschitz maps definable in Hensel minimal, non-trivially valued fields K of equicharacteristic zero. ...
The union of B is a finite union of 0-definable reparametrized open cells C i . ...
arXiv:2204.05900v2
fatcat:mwp2gowz3vh2jily3q5m6agb4m
Topological invariance of the combinatorial Euler characteristic of tame spaces
2011
Homology, Homotopy and Applications
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We prove the topological invariance of the combinatorial Euler characteristic with the help of a canonical, topologically defined stratification of tame spaces by locally compact, tame strata. ...
Any definable set X possesses finite decompositions into cylindrical cells and one can let eu S (X) = ∑ α∈cell(X) (−1) dim(α) , for any such decomposition. ...
Let W be definable, locally compact and stratified into definable sets. Let X be a union of strata. Define top(X) = union of {α ⊆ X | for all β ⊆ W and γ ⊆ X, if α β γ, then β ⊆ X}. ...
doi:10.4310/hha.2011.v13.n2.a11
fatcat:ifdbwoyv75hkbde2y6xbidouqm
O-minimalism
[article]
2011
arXiv
pre-print
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We propose a theory, DCTC (Definable Completeness/Type Completeness), that describes many properties of o-minimalistic structures (dimension theory, monotonicity, Hardy structures, quasi-cell decomposition ...
Failure of cell decomposition leads to the related notion of a tame structure, and we give a criterium for an o-minimalistic structure to be tame. ...
By Theorem 3.14, we can write Y as a disjoint union of open intervals and a discrete set D. ...
arXiv:1106.1196v1
fatcat:ik5lb72e6nhgtm3r746en3m7nq
O-MINIMALISM
2014
Journal of Symbolic Logic (JSL)
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As an application, we study certain analytic subsets, called Taylor sets. ...
, provided one replaces finiteness by discreteness in all of these. ...
(which in the o-minimal case does yield a finite cell decomposition), we can decompose each X i as a disjoint union of ∅-definable subsets X (e) i consisting of the union of all e-cells in a cell decomposition ...
doi:10.1017/jsl.2013.14
fatcat:hcrqbgyihfej3kt424wfrgeyuu
Definable Sets in Ordered Structures. II
1986
Transactions of the American Mathematical Society
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It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components." ...
Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets. ...
Roughly speaking, our proofs of the main theorems will go as follows: We will prove inductively, for each n: (i) any definable set X in Mn is a finite union of cells, (ii) any definable function /: Mn ...
doi:10.2307/2000053
fatcat:5juus5s6hnallbby6lqmlavmna
Locally o-minimal structures
2012
Journal of the Mathematical Society of Japan
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We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (R, +, <). ...
As in the o-minimal setting, we can define cells and cell decompositions of definable sets in the locally o-minimal setting, see [3] . ...
Recall that M is said to be o-minimal if every definable subset of M is a finite union of points and open intervals in M . ...
doi:10.2969/jmsj/06430783
fatcat:hx5zznke65faxpeityc6b6zzry
Definable sets in ordered structures. II
1986
Transactions of the American Mathematical Society
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It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components." ...
Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets. ...
Roughly speaking, our proofs of the main theorems will go as follows: We will prove inductively, for each n: (i) any definable set X in Mn is a finite union of cells, (ii) any definable function /: Mn ...
doi:10.1090/s0002-9947-1986-0833698-1
fatcat:3h5h3dhxczcsjofwd3rr5vyuwu
On groups and fields definable in o-minimal structures
1988
Journal of Pure and Applied Algebra
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The structure M = (M, <, R" R" .) is o-minimal if every definable set XC M is a finite union of intervals (a, b) and points. Let G be a group definable in M (i.e. ...
G is a definable subset of M" and the graph of multiplication is also definable). ...
First, any definable subset of V is a finite union of definable cells, each of which is locally closed (i.e. the intersection of an open set and a closed set) in Mk. ...
doi:10.1016/0022-4049(88)90125-9
fatcat:dmn7wheppncj5e33jzjqb4yvgq
Page 458 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 5, Issue 3
[page]
1954
American Mathematical Society. Proceedings of the American Mathematical Society
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It is easy to see that the closure of S,,—P 1 can be expressed as the union of a finite set oi, - - - , ¢, of elements of =. ...
An integer k and an (w—1)-cell J* are chosen as in case 2, and sets Sm(€) are defined as in case 2. We also choose 6 as in case 2. ...
Expansions of o-minimal structures by sparse sets
2001
Fundamenta Mathematicae
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The same holds true with "nowhere dense" replaced by any of "null" (in the sense of Lebesgue), "countable", "a finite union of discrete sets", or "discrete". ...
The same holds true with "nowhere dense" replaced (uniformly) by any of "null" (in the sense of Lebesgue), "countable", "a finite union of discrete sets", or "discrete". ...
By Theorem B, every subset of R definable in (R, +, ·, Fib) # , as well as every subset of R definable in (R, +, ·, ϕ Z ) # , is the union of an open set and finitely many discrete sets. ...
doi:10.4064/fm167-1-4
fatcat:52h3xzzfzjbtro5vwwgi4qrnoa
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