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Structure theorems for o-minimal expansions of groups

Mario J. Edmundo
2000 Annals of Pure and Applied Logic  
Let Rbe an o-minimal expanion of an ordered group (R, 0, 1, +, <) with distinguished positive element 1.  ...  endomorphism of the group (R, 0, +).  ...  suggesting some essential mathematical changes in early versions of the paper.  ... 
doi:10.1016/s0168-0072(99)00043-3 fatcat:z4ynxuexkbcopl5z5eknfvhdj4

Dp-minimal expansions of discrete ordered abelian groups [article]

Erik Walsberg
2020 arXiv   pre-print
If Z is a dp-minimal expansion of a discrete ordered abelian group (Z,<,+) and Z does not admit a nontrivial definable convex subgroup then Z is interdefinable with (Z,<,+) and (Z,<,+) is elementarily  ...  Two structures on a common domain M are interdefinable if they define the same subsets of M n for all n. An expansion of a structure M is proper if it is not interdefinable with M.  ...  Let (Z, O) be the expansion of Z by a unary predicate defining O. Fact 1.3 shows (Z, O) is dp-minimal.  ... 
arXiv:1903.06222v2 fatcat:d53yih5fpndwhjnmk6pkr4n5wq

Solvable Lie groups definable in o-minimal theories [article]

Annalisa Conversano, Alf Onshuus, Sergei Starchenko
2015 arXiv   pre-print
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.  ...  Groups definable in o-minimal structures. We refer to [18] for basics on o-minimal structures. In this paper we will need two particular o-minimal expansions of the real field.  ...  A significance of the above question is that for groups definable in o-minimal structures one can use the well developed and powerfull tools of o-minimality (see [17] and [20] for more details on o-minimality  ... 
arXiv:1506.08131v1 fatcat:27r7fga5ezh5lngq4ywlry4noe

Externally definable quotients and NIP expansions of the real ordered additive group [article]

Erik Walsberg
2020 arXiv   pre-print
Let R be an NIP expansion of (R,<,+) by closed subsets of R^n and continuous functions f : R^m →R^n. Then R is generically locally o-minimal.  ...  This follows from a more general theorem on NIP expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable  ...  Local and generic local o-minimality. Theorem 2.13 gives several equivalent definitions of local o-minimality for expansions of (R, <, +). Locally o-minimal structures are studied in [80, 51] .  ... 
arXiv:1910.10572v3 fatcat:i6ui3zglxvalhh3l4dpvgktdqi

Definable Tietze extension property in o-minimal expansion of ordered group [article]

Masato Fujita
2021 arXiv   pre-print
The following two assertions are equivalent for an o-minimal expansion of an ordered group ℳ=(M,<,+,0,...). There exists a definable bijection between a bounded interval and an unbounded interval.  ...  Any definable continuous function f:A → M defined on a definable closed subset of M^n has a definable continuous extension F:M^n → M.  ...  Theorem 1 . 2 . 12 Consider an o-minimal expansion M = (M, <, +, 0, . . .) of an ordered group.  ... 
arXiv:2011.02922v2 fatcat:bcrbttoa2zczlip5s7fgajhlqu

On O-Minimal Expansions of (Q,<,+,0) [article]

Pablo Cubides Kovacsics, Françoise Delon
2017 arXiv   pre-print
Let f:Q→Q be a function definable in an o-minimal expansion of (Q,<,+,0). We show that f is eventually linear. In addition, we show that this holds in every elementary equivalent structure.  ...  We would also like to thank Antoine Chambert-Loir, Marc Hindry and César Martínez-Metzmeier for interesting discussions and for their help concerning the use of Hilbert irreducibility theorem.  ...  We would like to thank Alex Wilkie for pointing us out several results in the field, including his answer to Steinhorn's question.  ... 
arXiv:1704.03050v2 fatcat:tpwk23cornfzpps52gp3jw2iwm

Dp and other minimalities [article]

Pierre Simon, Erik Walsberg
2022 arXiv   pre-print
A first order expansion of (ℝ,+,<) is dp-minimal if and only if it is o-minimal.  ...  In the last section we give a canonical correspondence between dp-minimal expansions of (ℚ,+,<) and o-minimal expansionsof (ℝ,+,<) such that (ℛ,ℚ) is a "dense pair".  ...  We now suppose that X ⊆ K n is definable in K Sh and show that X is definable in the structure induced on K by K K K. By dp-minimality of K Sh and Fact 7.8 we may suppose that X is closed in K n .  ... 
arXiv:1909.05399v3 fatcat:umpodywqwvfavjdpsu5ahoals4

Generalizing a theorem of Bès and Choffrut [article]

Erik Walsberg
2020 arXiv   pre-print
We prove a generalization: if R is an o-minimal expansion of (R,<,+) by bounded subsets of Euclidean space then there are no intermediate structures between R and (R,Z).  ...  It follows there are no intermediate structures between (R,<,+,sin|_[0,2π]) and (R,<,+,sin).  ...  Expansions of o-minimal structures by Z We survey structures of the form (R, Z) where R is o-minimal.  ... 
arXiv:2002.10508v1 fatcat:m2vm3h5rardknat25hg4oilkva

Page 6067 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
and Poizat showing that any expansion of an o-minimal model by predicates for convex subsets is weakly o-minimal.  ...  For instance, let us mention the expansion of the (o-minimal) ordered field of real algebraic numbers by any interval (a, b) where a < b are transcen- dental.  ... 

Almost o-minimal structures and 𝔛-structures [article]

Masato Fujita
2021 arXiv   pre-print
Another contribution of this paper is a uniform local definable cell decomposition theorem for almost o-minimal expansions of ordered groups ℳ=(M,<,0,+,...).  ...  We demonstrate that an 𝔛-expansion of an ordered divisible abelian group always contains an o-minimal expansion of an ordered group such that all bounded 𝔛-definable sets are definable in the structure  ...  Take the o-minimal expansion of an ordered group R given in Theorem 2.13.  ... 
arXiv:2104.01312v1 fatcat:puwzoagquffhpphd2dsf5c7dbm

An introduction to o-minimal structures [article]

Mario J. Edmundo
2000 arXiv   pre-print
We omit here any reference to the pure model theory of o-minimal structures and to the theory of groups and rings definable in o-minimal structures.  ...  In these notes we try to show that this is in fact the case by presenting several examples of o-minimal structures and by listing some geometric properties of sets and maps definable in o-minimal structures  ...  The models of RCF are called real closed fields and they are examples of o-minimal structures (see the definition below). O-minimal structures O-minimal structures.  ... 
arXiv:math/0012051v2 fatcat:5gavj6otorbx5darfs7q6tokwq

Dp-minimal expansions of (Z,+) via dense pairs via Mordell-Lang [article]

Erik Walsberg
2020 arXiv   pre-print
We canonically associate an o-minimal expansion R of (R,+,×), an R-definable circle group H, and a character Z→H to a "non-modular" dp-minimal expansion of (Z,+,S).  ...  This is a contribution to the classification problem for dp-minimal expansions of (Z,+). Let S be a dense cyclic group order on (Z,+).  ...  This will reduce the question "what are the dp-minimal expansions of (Z, +, S) to "for which o-minimal expansions C of (R/Z, +, C) is the structure induced on Z by C and ψ dp-minimal"?  ... 
arXiv:2004.06847v1 fatcat:enhxbh63d5g53pw6yuokukfh5i

Page 5752 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews  
Let R be a polynomially bounded o-minimal expansion of the field of reals.  ...  The author gives a characterization of one-dimensional, definably connected groups definable in an o-minimal expansion of a real closed field.  ... 

ON DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NONVALUATIONAL STRUCTURES

PANTELIS E. ELEFTHERIOU, ASSAF HASSON, GIL KEREN
2017 Journal of Symbolic Logic (JSL)  
We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions.  ...  We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.  ...  We also thank Philipp Hieronymi and Moshe Kamensky for their important feedback, and the referee for many useful suggestions. 1 This means that the order and the group structure are preserved in the  ... 
doi:10.1017/jsl.2017.28 fatcat:5axxbqacwrdl5mxdenenvzi5wq

Tameness of definably complete locally o-minimal structures and definable bounded multiplication [article]

Masato Fujita and Tomohiro Kawakami and Wataru Komine
2021 arXiv   pre-print
Similarly to o-minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze extension theorem and affiness of psudo-definable spaces hold true for such structures under the extra assumption that  ...  We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure.  ...  Acknowledgment MF would like to express his gratitude to the department of mathematics, Hiroshima university for its provision of references. WK appreciates Akito Tsuboi and Kota Takeuchi.  ... 
arXiv:2110.15613v1 fatcat:izkofm2s35gyvagwchafb3gvla
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