The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of …
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its …
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.
Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version …
Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.
We prove that in NTP_2 theories if p is a dependent type with dp-rank >= \kappa, then this can be witnessed by indiscernible sequences of tuples satisfying p. If p …
We prove that in NTP_2 theories if p is a dependent type with dp-rank >= \kappa, then this can be witnessed by indiscernible sequences of tuples satisfying p. If p has dp-rank infinity, then this can be witnessed by singletons (in any theory).
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing …
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing …
In this paper, we build a dimension theory related to Shelah's 2-rank, dp-rank, and o-minimal dimension. We call this dimension op-dimension. We exhibit the notion of the n-multi-order property, generalizing the order property, and use this to create op-rank, which generalizes 2-rank. From this we build op-dimension. We show that op-dimension bounds dp-rank, that op-dimension is sub-additive, and op-dimension generalizes o-minimal dimension in o-minimal theories.
Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. …
Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called $m$-determinacy and show that theories of distality rank $m$ require certain products to be $m$-determined. Furthermore, for NIP theories, this behavior characterizes $m$-distality. If we narrow the scope to stable theories, we observe that $m$-distality can be characterized by the maximum cycle size found in the forking "geometry," so it coincides with $(m-1)$-triviality. On a broader scale, we see that $m$-distality is a strengthening of Saharon Shelah's notion of $m$-dependence.
Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. …
Building on Pierre Simon's notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called $m$-determinacy and show that theories of distality rank $m$ require certain products to be $m$-determined. Furthermore, for NIP theories, this behavior characterizes $m$-distality. If we narrow the scope to stable theories, we observe that $m$-distality can be characterized by the maximum cycle size found in the forking geometry, so it coincides with $(m-1)$-triviality. On a broader scale, we see that $m$-distality is a strengthening of Saharon Shelah's notion of $m$-dependence.
For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, …
For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.
Abstract Building on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank m such that $1\leq m \leq …
Abstract Building on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank m such that $1\leq m \leq \omega $ . For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called m -determinacy and show that theories of distality rank m require certain products to be m -determined. Furthermore, for NIP theories, this behavior characterizes m -distality. If we narrow the scope to stable theories, we observe that m -distality can be characterized by the maximum cycle size found in the forking “geometry,” so it coincides with $(m-1)$ -triviality. On a broader scale, we see that m -distality is a strengthening of Saharon Shelah’s notion of m -dependence.
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, …
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, …
In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula---plus a …
We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula---plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-companion is strongly minimal. We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension.
We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a …
We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-companion is strongly minimal. We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension.
In this paper, we introduce the notion of K-rank, where K is an strong amalgamation Fraisse class. Roughly speaking, the K-rank of a partial type is the number of "copies" …
In this paper, we introduce the notion of K-rank, where K is an strong amalgamation Fraisse class. Roughly speaking, the K-rank of a partial type is the number of "copies" of K that can be "independently coded" inside of the type. We study K-rank for specific examples of K, including linear orders, equivalence relations, and graphs. We discuss the relationship of K-rank to other ranks in model theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work).
We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question …
We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question for dp-rank. We show that if $B$ contains a Morley sequence $I$ generated by $p$ over $A$, then $p\restriction AI$ is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of "trees whose open cones are models of some theory" and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.
Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types …
Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types for stable formulas, one develops the powerful tools of stability theory, such as canonical bases, a nice forking calculus, and stable embeddability. When one passes to the class of dependent formulas, this notion of definability of types is lost. However, as this dissertation shows, we can recover suitable alternatives to definability of types for some dependent theories. Using these alternatives, we can recover some of the power of stability theory. One alternative is uniform definability of types over finite sets (UDTFS). We show that all formulas in dp-minimal theories have UDTFS, as well as formulas with VC-density < 2. We also show that certain Henselian valued fields have UDTFS. Another alternative is isolated extensions. We show that dependent formulas are characterized by the existence of isolated extensions, and show how this gives a weak stable embeddability result. We also explore the idea of UDTFS rank and show how it relates to VC-density. Finally, we use the machinery developed in this dissertation to show that VCminimal theories satisfy the Kueker Conjecture. ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES