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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

The proof is complicated by the fact that a

arXiv:1910.13504v1
fatcat:xsejcsy2m5eu5h45koqmyhb5dq
*definable**open**set*need not be a*union**of**finitely*many*open**cells*. ... U is a*finite**union**of*regular*open*R-*definable**sets*. ...##
###
Structures having o-minimal open core

2009
*
Transactions of the American Mathematical Society
*

By the

doi:10.1090/s0002-9947-09-04908-3
fatcat:gqydu6uuezgandlqtgmd7r4cuu
*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. ...##
###
Almost o-minimal structures and 𝔛-structures
[article]

2021
*
arXiv
*
pre-print

We introduce the notion

arXiv:2104.01312v1
fatcat:puwzoagquffhpphd2dsf5c7dbm
*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. ...##
###
Locally o-minimal structures and structures with locally o-minimal open core

2013
*
Annals of Pure and Applied Logic
*

We study first-order expansions

doi:10.1016/j.apal.2012.10.002
fatcat:fmwztdzvvva35prhg3qunus77i
*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*. ...##
###
A theorem about topological $n$-cells

1954
*
Proceedings of the American Mathematical Society
*

*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. ...

##
###
Extension of Lipschitz maps definable in Hensel minimal structures
[article]

2022
*
arXiv
*
pre-print

It may be regarded

arXiv:2204.05900v2
fatcat:mwp2gowz3vh2jily3q5m6agb4m
*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 . ...##
###
Topological invariance of the combinatorial Euler characteristic of tame spaces

2011
*
Homology, Homotopy and Applications
*

We prove the topological invariance

doi:10.4310/hha.2011.v13.n2.a11
fatcat:ifdbwoyv75hkbde2y6xbidouqm
*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}. ...##
###
O-minimalism
[article]

2011
*
arXiv
*
pre-print

We propose a theory, DCTC (

arXiv:1106.1196v1
fatcat:ik5lb72e6nhgtm3r746en3m7nq
*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. ...##
###
O-MINIMALISM

2014
*
Journal of Symbolic Logic (JSL)
*

*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 ...

##
###
Definable Sets in Ordered Structures. II

1986
*
Transactions of the American Mathematical Society
*

It is simultaneously proved that if M is 0minimal, then every

doi:10.2307/2000053
fatcat:5juus5s6hnallbby6lqmlavmna
*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 ...##
###
Locally o-minimal structures

2012
*
Journal of the Mathematical Society of Japan
*

We first give a characterization

doi:10.2969/jmsj/06430783
fatcat:hx5zznke65faxpeityc6b6zzry
*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 . ...##
###
Definable sets in ordered structures. II

1986
*
Transactions of the American Mathematical Society
*

It is simultaneously proved that if M is 0minimal, then every

doi:10.1090/s0002-9947-1986-0833698-1
fatcat:3h5h3dhxczcsjofwd3rr5vyuwu
*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 ...##
###
On groups and fields definable in o-minimal structures

1988
*
Journal of Pure and Applied Algebra
*

The structure M = (M, <, R" R" .) is o-minimal if every

doi:10.1016/0022-4049(88)90125-9
fatcat:dmn7wheppncj5e33jzjqb4yvgq
*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. ...##
###
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
*

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
*

The same holds true with "nowhere dense" replaced by any

doi:10.4064/fm167-1-4
fatcat:52h3xzzfzjbtro5vwwgi4qrnoa
*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*. ...
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