Structures having o-minimal open core

Dolich, Alfred; Miller, Chris; Steinhorn, Charles

doi:10.1090/s0002-9947-09-04908-3

Cited by 70 publications

(162 citation statements)

References 32 publications

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“…This condition plays a crucial rôle in [6], and provides a hint for relaxing the assumption of o-minimality of T in [6, 6.1].…”

Section: No New Open Setsmentioning

confidence: 99%

“…To show that X has no interior is thus the same as ifT c were ominimal, so the rest of the proof goes through unchanged. The proof for o-minimal T is considerably easier, and can be obtained by appropriately modifying arguments found in [6] or [10].…”

Section: No New Open Setsmentioning

confidence: 99%

“…As in [6], we say that a structure M has uniform finiteness (UF, for short) if for every m ∈ N and A ⊆ M m × M definable in M there exists N ∈ N such that for every x ∈ M m the set { y ∈ M : (x, y) ∈ A } either is infinite or contains at most N elements. We say that a theory T has UF if every model of T has UF.…”

Section: Introductionmentioning

confidence: 99%

“…The open core of a structure M endowed with a definable topology is the structure on the underlying set of M generated by the collection of all open sets that are definable in M. A weaker version of Theorem 1.9 was established in [6] under the additional assumption that no model of T defines a unary function having a somewhere-dense graph-see the beginning of the proof of [6, 6.1]-but this extra condition is rather strong and is easily seen to fail in many concrete cases of interest; below are two (each already mentioned earlier): Example 1.12 ( [7]). If R is an o-minimal expansion of a densely ordered group and E ⊆ R is dense and definably independent, then Th( R, E ) defines no new open sets over Th(R), and every model of Th( R, E ) defines a unary function whose graph is dense.…”

Section: Introductionmentioning

confidence: 99%

“…Before proceeding to proofs, we digress briefly to expand on an issue raised in [6]. If M has UF, then Th(M) has generic extensions by Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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Extensions of ordered theories by generic predicates

Dolich

Miller²,

Steinhorn³

2013

J. symb. log.

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Given a theory T extending that of dense linear orders without endpoints (DLO), in a language ℒ ⊇ {<}, we are interested in extensions T′ of T in languages extending ℒ by unary relation symbols that are each interpreted in models of T′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namely T = Th(〈ℝ, <, +, ·〉) and T′ = Th(〈ℝ, <, +, · ℚ 〉). Recall that T is o-minimal, and so every open set definable in any model of T has only finitely many definably connected components. But it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models of T are essentially as simple as possible, while T′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.In contrast to the preceding example, if ℝalg is the set of real algebraic numbers and T′ Th(〈ℝ, <, +, ·, 〈alg〉), then no model of T′ defines any open set (of any arity) that is not definable in the underlying model of T.

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“…This condition plays a crucial rôle in [6], and provides a hint for relaxing the assumption of o-minimality of T in [6, 6.1].…”

Section: No New Open Setsmentioning

confidence: 99%

Section: No New Open Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Before proceeding to proofs, we digress briefly to expand on an issue raised in [6]. If M has UF, then Th(M) has generic extensions by Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Extensions of ordered theories by generic predicates

Dolich

Miller²,

Steinhorn³

2013

J. symb. log.

Self Cite

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Computable axiomatizability of elementary classes

Sinclair

2016

Mathematical Logic Qtrly

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The goal of this paper is to generalise Alex Rennet's proof of the non-axiomatizability of the class of pseudo-o-minimal structures. Rennet showed that if L is an expansion of the language of ordered fields and K is the class of pseudo-o-minimal L-structures (L-structures elementarily equivalent to an ultraproduct of o-minimal structures) then K is not computably axiomatizable. We give a general version of this theorem, and apply it to several classes of topological structures.

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Distal and non‐distal behavior in pairs

Nell

2019

Mathematical Logic Qtrly

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The aim of this work is an analysis of distal and non-distal behavior in dense pairs of o-minimal structures. A characterization of distal types is given through orthogonality to a generic type in M eq , non-distality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.Definition 2.1 Let M |= T , s = (s 1 , . . . , s n ) and t = (t 1 , . . . , t m ) be tuples of sorts, and A ⊆ M s be definable. Then for any definable or type-definable S ⊆ M t , we say S is A-small if there is ∈ N and a definable f :We now remind the reader of certain definitions involving types that appear often in the context of NIP theories. Definition 2.2 Let t = (t 1 , . . . , t n ) be a tuple of sorts, and p(x) ∈ S t (U t ). We say p(x) is generically stable if there is a small A ⊂ U such that

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Structures having o-minimal open core

Abstract: Abstract. The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets.

Cited by 70 publications

References 32 publications

Extensions of ordered theories by generic predicates

Extensions of ordered theories by generic predicates

Computable axiomatizability of elementary classes

Distal and non‐distal behavior in pairs

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