NIP Theories Research Articles

Density of compressible types and some consequences

We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we obtain explicit uniform honest definitions for NIP formulas (answering a question of Eshel and the second author), and build compressible models in countable NIP theories.

ON LARGE EXTERNALLY DEFINABLE SETS IN NIP

Abstract We study cofinal systems of finite subsets of $\omega _1$ . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

A definable (p,q)-theorem for NIP theories

We prove a definable version of Matoušek's (p,q)-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version.The proof builds on a proof of Boxall and Kestner who proved this theorem in the distal case. It utilizes the notion of locally compressible types which appeared in the work of the author with Bays and Simon.

MAXIMAL STABLE QUOTIENTS OF INVARIANT TYPES IN NIP THEORIES

Abstract For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$ ) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3].

On minimal flows and definable amenability in some distal NIP theories

We study the definable topological dynamics (G(M),SG(M)) of a definable group acting on its type space, where M is either an o-minimal structure or a p-adically closed field, and G a definably amenable group. We focus on the problem raised in [13] of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when G has boundedly many global weakly generic types. We also give two “minimal counterexamples” where G has unboundedly many global weakly generic types, extending the main results in [21] to a more general context.

Type-definable NIP fields are Artin–Schreier closed

Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p \gt 0$, then $K$ is Artin–Schreier closed (it has no Artin–Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension o

DISTALITY RANK

AbstractBuilding 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.

On Stable Quotients

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic.

Definable convolution and idempotent Keisler measures

We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of definable and finitely satisfiable (or just definable, assuming NIP) measures are nice semigroups, and classify idempotent measures in stable groups as invariant measures on type-definable subgroups. We establish left-continuity of the convolution map in NIP theories, and use it to show that the convolution semigroup on finitely satisfiable measures is isomorphic to a particular Ellis semigroup in this context.

BOOLEAN TYPES IN DEPENDENT THEORIES

AbstractThe notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

Linear orders in NIP structures

We show that every unstable NIP theory admits a ⋁-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is ω-categorical, then it interprets an infinite linear order. This partially answers a longstanding open question.

ASSOCIATIVITY OF THE MORLEY PRODUCT OF INVARIANT MEASURES IN NIP THEORIES

AbstractIn light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories.

Criteria for exact saturation and singular compactness

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.

Hereditary G-compactness

We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable (and by a result of Simon, this holds unconditionally for $\aleph_0$-categorical theories). We show that if $G$ is definable over $A$ in a hereditarily G-compact theory, then $G^{00}_A=G^{000}_A$. We also include a brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some (classes of) theories.

Ramsey transfer to semi-retractions

We introduce the notion of a semi-retraction. Given two structures A and B, A is a semi-retraction of B if there exist quantifier-free type respecting maps f:B→A and g:A→B such that f∘g is an embedding. We say that a structure has the Ramsey property if its age does. Given two locally finite ordered structures A and B, if A is a semi-retraction of B and B has the Ramsey property, then A also has the Ramsey property. We introduce notation for what we call semi-direct product structures, after the group construction known to preserve the Ramsey property [11]. We introduce the notion of a color-homogenizing map, and use this notion to give a finitary argument that the semi-direct product structure of ordered relational structures with the Ramsey property must also have the Ramsey property. Finally, we characterize NIP theories using a generalized indiscernible sequence indexed by a semi-direct product structure.

Type decomposition in NIP theories

A first order theory is NIP if all definable families of subsets have finite VC-dimension. We provide a justification for the intuition that NIP structures should be a combination of stable and order-like components. More precisely, we prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and an order-like quotient.

Boundedness and absoluteness of some dynamical invariants in model theory

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text].

Remarks on generic stability in independent theories

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs.

Model theory, Keisler measures, and groups - Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596. - Ehud Hrushovski and Anand Pillay, On NIP and invariant measures. Journal of the European Mathematical Society, vol.13 (2011), no. 4, pp. 1005–1061. - Ehud Hrushovski, Anand Pillay,

Model theory, Keisler measures, and groups - Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596. - Ehud Hrushovski and Anand Pillay, On NIP and invariant measures. Journal of the European Mathematical Society, vol.13 (2011), no. 4, pp. 1005–1061. - Ehud Hrushovski, Anand Pillay,

ON MAXIMAL STABLE QUOTIENTS OF DEFINABLE GROUPS IN NIP THEORIES

AbstractFor G a group definable in a saturated model of a NIP theory T, we prove that there is a smallest type-definable subgroup H of G such that the quotient G / H is stable. This generalizes the existence of G00, the smallest type-definable subgroup of G of bounded index.