Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences …
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
Abstract We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example …
Abstract We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.
We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated …
We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T-convex theory.
Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación …
Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación dada en la Definición 1.3). También demostramos que si tenemos þ-división podemos lograr división fuerte sobre una base que pertenece a la clausura algebraica del conjunto parámetro.
Abstract We let R be an o-minimal expansion of a field, V a convex subring, and ( R 0 , V 0 ) an elementary substructure of ( R , …
Abstract We let R be an o-minimal expansion of a field, V a convex subring, and ( R 0 , V 0 ) an elementary substructure of ( R , V ). Our main result is that ( R , V ) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R , provided that k R (the residue field with structure induced from R ) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k with structure induced from ( R , V ). We also give a criterion for a superstructure of ( R , V ) being an elementary extension of ( R , V ).
We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of …
We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for classical real rosy theories.
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable …
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of …
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).
We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in …
We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed.
We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose …
We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.
Let R be an o-minimal expansion of a group in a language in which Th(R) eliminates quantifiers, and let C be a predicate for a valuational cut in R. We …
Let R be an o-minimal expansion of a group in a language in which Th(R) eliminates quantifiers, and let C be a predicate for a valuational cut in R. We identify a condition that implies quantifier elimination for Th(R,C) in the language of R expanded by C and a small number of constants, and which, in turn, is implied by Th(R,C) having quantifier elimination and being universally axiomatizable. The condition applies for example in the case when C is a convex subring of an o-minimal field R and its residue field is o-minimal.
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of …
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).
Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We …
Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We identify a condition that implies quantifier elimination for $\textrm{Th}(R,C)$ in the language of $R$ expanded by $C$ and a small number of constants, and which, in turn, is implied by $\textrm{Th}(R,C)$ having quantifier elimination and being universally axiomatizable. The condition applies for example in the case when $C$ is a convex subring of an o-minimal field $R$ and its residue field is o-minimal.
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable …
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable …
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.
Let R be an o-minimal expansion of a group in a language in which Th(R) eliminates quantifiers, and let C be a predicate for a valuational cut in R. We …
Let R be an o-minimal expansion of a group in a language in which Th(R) eliminates quantifiers, and let C be a predicate for a valuational cut in R. We identify a condition that implies quantifier elimination for Th(R,C) in the language of R expanded by C and a small number of constants, and which, in turn, is implied by Th(R,C) having quantifier elimination and being universally axiomatizable. The condition applies for example in the case when C is a convex subring of an o-minimal field R and its residue field is o-minimal.
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable …
We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort.
Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We …
Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We identify a condition that implies quantifier elimination for $\textrm{Th}(R,C)$ in the language of $R$ expanded by $C$ and a small number of constants, and which, in turn, is implied by $\textrm{Th}(R,C)$ having quantifier elimination and being universally axiomatizable. The condition applies for example in the case when $C$ is a convex subring of an o-minimal field $R$ and its residue field is o-minimal.
We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated …
We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T-convex theory.
Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación …
Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación dada en la Definición 1.3). También demostramos que si tenemos þ-división podemos lograr división fuerte sobre una base que pertenece a la clausura algebraica del conjunto parámetro.
Abstract We let R be an o-minimal expansion of a field, V a convex subring, and ( R 0 , V 0 ) an elementary substructure of ( R , …
Abstract We let R be an o-minimal expansion of a field, V a convex subring, and ( R 0 , V 0 ) an elementary substructure of ( R , V ). Our main result is that ( R , V ) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R , provided that k R (the residue field with structure induced from R ) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k with structure induced from ( R , V ). We also give a criterion for a superstructure of ( R , V ) being an elementary extension of ( R , V ).
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of …
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).
Abstract We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example …
Abstract We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of …
We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).
We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of …
We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for classical real rosy theories.
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences …
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in …
We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed.
We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose …
We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.
A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of …
A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of …
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences …
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
Abstract Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that …
Abstract Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T . We give a quantifier elimination relative to T for the theory of pairs (ℛ, V ) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are …
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G …
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].
In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let $R$ be an o-minimal field with …
In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let $R$ be an o-minimal field with a proper convex subring $V$, and let $\mathop{\rm st}: V \to \boldsymbol k$
This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and …
This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.
Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable …
Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M , generalizing a result of Dolich on o-minimal theories in [4].
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain …
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating …
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of …
(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of having proved that theorem.
I solve here some problems left open in “ T -convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular, T …
I solve here some problems left open in “ T -convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular, T will denote a complete o -minimal theory extending RCF, the theory of real closed fields. Let ( , V ) ⊨ T convex , let = V /m( V ) be the residue field, with residue class map x ↦ : V ↦ , and let υ: → Γ be the associated valuation. “Definable” will mean “definable with parameters”. The main goal of this article is to determine the structure induced by ( , V ) on its residue field and on its value group Γ . In [9] we expanded the ordered field to a model of T as follows. Take a tame elementary substructure ′ of such that R ′ ⊆ V and R′ maps bijectively onto under the residue class map, and make this bijection into an isomorphism ′ ≌ of T -models. (We showed such ′ exists, and that this gives an expansion of to a T -model that is independent of the choice of ′.).
Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples …
Let M = 〈M, <, …〉 be alinearly ordered structure. We define M to be o-minimal if every definable subset of M is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal M can induce on a neighbourhood of any a in M. Roughly said, one of the following holds: (i) a is trivial (technical term), or (ii) a has a convex neighbourhood on which M induces the structure of an ordered vector space, or (iii) a is contained in an open interval on which M induces the structure of an expansion of a real closed field. The proof uses 'geometric calculus' which allows one to recover a differentiable structure by purely geometric methods. 1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.
Abstract Let D be a strongly minimal set in the language L , and D ′ ⊃ D an elementary extension with infinite dimension over D . Add to L …
Abstract Let D be a strongly minimal set in the language L , and D ′ ⊃ D an elementary extension with infinite dimension over D . Add to L a unary predicate symbol D and let T ′ be the theory of the structure ( D ′, D ), where D interprets the predicate D . It is known that T ′ is ω -stable. We prove Theorem A. If D is not locally modular, then T ′ has Morley rank ω . We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a , b ∈ D and closed X ⊂ D , a ∈ cl( Xb ) ⇒ there is a Y ⊂ X with a ∈ cl( Yb ) and ∣ Y ∣ ≤ k . Using Theorem A, we prove Theorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective . The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B. Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D 1 of D such that the theory of the pair (D 1 , D) is ω 1 -categorical. Then D is locally modular .
Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for …
Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result: Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that : (1) The symbol “∣” is interpreted as the honest divisibility relation : (2) The following divisibility property holds in T : If T admits q.e. in ℒ, then T = RCVR. We do not know at present whether the restriction imposed by condition (2) can be weakened. The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings , for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings: Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent : (a) A is a convexly ordered valuation ring . (b) Every ideal (or, equivalently, principal ideal) is convex in A . (c) A is a valuation ring convex in its field of fractions quot( A ). (d) A is a valuation ring and its maximal ideal M A is convex (in A or, equivalently, in quot (A)) . (e) A is a valuation ring and its maximal ideal is bounded by ± 1.
In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical …
In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine …
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model …
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p -adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.
Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space …
Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space $\ell^2({\mathbb N})$. The proof is based on a lem
Let K be the (real closed) field of Puiseux series in t over R endowed with the natural linear order. Then the elements of the formal power series rings [[ξ1,…,ξn]] …
Let K be the (real closed) field of Puiseux series in t over R endowed with the natural linear order. Then the elements of the formal power series rings [[ξ1,…,ξn]] converge t-adically on [−t,t]n, and hence define functions [−t,t]n → K. Let L be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.15, K is o-minimal in L. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis. 2000 Mathematics Subject Classification 03C64, 32P05, 32B05, 32B20, 03C10 (primary), 03C98, 03C60, 14P15 (secondary).
Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. …
Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T . canonical base of an amalgamation class is the union of names of ψ -definitions of , ψ ranging over stationary L -formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.
We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones …
We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.
The notion of a randomization of a first-order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new …
The notion of a randomization of a first-order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new structure whose elements are random elements of the original first-order structure. In this paper we treat randomizations as continuous structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the earlier results show that the randomization of a complete first-order theory is a complete theory in continuous logic that admits elimination of quantifiers and has a natural set of axioms. We show that the randomization operation preserves the properties of being omega-categorical, omega-stable, and stable.
A metric structure is a many-sorted structure in which each sort is a complete metric space of finite diameter. Additionally, the structure consists of some distinguished elements as well as …
A metric structure is a many-sorted structure in which each sort is a complete metric space of finite diameter. Additionally, the structure consists of some distinguished elements as well as some functions (of several variables) (a) between sorts and (b) from sorts to bounded subsets of ℝ, and these functions are all required to be uniformly continuous. Examples arise throughout mathematics, especially in analysis and geometry. They include metric spaces themselves, measure algebras, asymptotic cones of finitely generated groups, and structures based on Banach spaces (where one takes the sorts to be balls), including Banach lattices, C*-algebras, etc.
We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model …
We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model complete and o-minimal structure which is exponentially bounded, and in which the Gamma function on the positive real line is definable. 2000 Mathematics Subject Classification: primary 03C10, 32B05, 32B20; secondary, 26E05.
We prove the following two theorems on embedded o-minimal structures: Theorem 1. Let ℳ ≺ 풩 be o-minimal structures and let ℳ* be the expansion of ℳ by all traces …
We prove the following two theorems on embedded o-minimal structures: Theorem 1. Let ℳ ≺ 풩 be o-minimal structures and let ℳ* be the expansion of ℳ by all traces in M of 1-variable formulas in 풩, that is all sets of the form φ(M, ā) for ā ⊆ N and φ(x, ȳ) ∈ ℒ(풩). Then, for any N-formula ψ(x1, …, xk), the set ψ(Mk) is ℳ*-definable. Theorem 2. Let 풩 be an ω1-saturated structure and let S be a sort in 풩eq. Let 풮 be the 풩-induced structure on S and assume that 풮 is o-minimal. Then 풮 is stably embedded.
The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the …
The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
We give an exposition of material from \[1], \[2] and \[3], regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. …
We give an exposition of material from \[1], \[2] and \[3], regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of generic reparametrization. I also try to bring out the relation to the geometry of \[3] - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
Let $R$ be an o-minimal field and $V$ a proper convex subring with residue field $\boldsymbol{k}$ and standard part (residue) map $\mathop{\rm st} \colon V\to \boldsymbol{k}$. Let $\boldsymbol{k}_{\rm ind}$ be …
Let $R$ be an o-minimal field and $V$ a proper convex subring with residue field $\boldsymbol{k}$ and standard part (residue) map $\mathop{\rm st} \colon V\to \boldsymbol{k}$. Let $\boldsymbol{k}_{\rm ind}$ be the expansion of $\boldsymbol{k}$ by the sta