Monotonicity and dimension in dpinp-minimal groups · NIP and inp-minimalityUnary setsUnary functionsHigher dimensions Monotonicity and dimension in dpinp-minimal groups John Goodrick

NIP and inp-minimality Unary sets Unary functions Higher dimensions

Monotonicity and dimension in dp inp-minimalgroups

John Goodrick

University of Maryland, College Park

University of LeedsJune 30, 2008

John Goodrick University of Maryland, College Park

Monotonicity and dimension in dp inp-minimal groups


Introduction

This talk will focus on definable sets in inp-minimal denselyordered abelian groups.

We will start with definable unary sets and explain why thesegroups are close to o-minimal.

Then we will discuss definable unary functions, culminating ina “Monotonicity Theorem:” Any definable unary function isthe union of finitely many continuous, locally monotonicfunctions.

Finally, we will discuss definable sets and functions in higherdimensions, where we have a few results (e.g. “No DenseGraphs”) but many conjectures.




Introduction








Introduction








Introduction








Inp-minimality: the definition

Let x be a single variable in the home sort (i.e. not an imaginary).

Definition

A theory is inp-minimal if there does not exist an inp-pattern oflength 2 for x = x , i.e. bdn(x = x) = 1.(See picture on blackboard.)

Definition

dp-minimal = inp-minimal + NIP






Definition


Definition







Definition


Definition





First example

Let T = Th(Q;<,+,P), where P is a dense, codense divisiblesubgroup. Then T is dependent, strong, and has o-minimal opencore.

But it is not inp-minimal.For appropriate choices of ai , bi , and ci , the following is aninp-pattern:

a0 < x < b0 a1 < x < b1 a2 < x < b2 . . . (2− inconsistent)P(x − c0) P(x − c1) P(x − c2) . . . (2− inconsistent)




First example

Let T = Th(Q;<,+,P), where P is a dense, codense divisiblesubgroup. Then T is dependent, strong, and has o-minimal opencore.

But it is not inp-minimal.For appropriate choices of ai , bi , and ci , the following is aninp-pattern:

a0 < x < b0 a1 < x < b1 a2 < x < b2 . . . (2− inconsistent)P(x − c0) P(x − c1) P(x − c2) . . . (2− inconsistent)




The bigger picture

See the blackboard for the bigger picture.




Why dp-minimality?

So why study dp-minimal (and inp-minimal) theories?

First reason: dp-minimal theories are the “tamest” “natural”class of NIP theories defined so far, and maybe there is hopeof understanding them.

Second reason: inp-minimality generalizes both strongminimality and o-minimality, so perhaps we can develop aunified dimension theory in this setting.




Why dp-minimality?







Why dp-minimality?







Why divisible ordered groups?

Theorem

(Onshuus, Usvyatsov) If T is stable, then T is inp-minimal if andonly if every 1-type has weight 1.

Definition

A densely ordered group (or DOG ) is an ordered abelian groupwith dense ordering (and maybe additional definable structure).

Question

Which DOGs are inp-minimal?

The ordered group imposes a homogeneity on the structure

which is useful for understanding definable sets.





Theorem


Definition


Question








Theorem


Definition


Question








Theorem


Definition


Question







A finiteness lemma

Lemma

(G.) If (M;<,+, . . .) is an inp-minimal densely ordered group,X ⊆ M is a definable, nowhere dense set, then X is finite.




A finiteness lemma

Lemma


For instance, to see that the set Z cannot be defined in aninp-minimal expansion (R;<,+, . . .) of the reals:




A finiteness lemma

Lemma



Pick elements ai , bi ∈ R such that

a0 < 0 < b0 < a1 < 1 < b1 < a2 < 2 < b2 < . . .

and bi − i = i − ai = 13 .

Pick a positive number ε0 <13 , and pick a decreasing sequence

ε0 > ε1 > ε2 > . . . > 0.




A finiteness lemma


Pick elements ai , bi ∈ R such that

a0 < 0 < b0 < a1 < 1 < b1 < a2 < 2 < b2 < . . .

and bi − i = i − ai = 13 .

Pick a positive number ε0 <13 , and pick a decreasing sequence

ε0 > ε1 > ε2 > . . . > 0.

If Z were definable, we would get a contradiction to inp-minimalityvia:

ϕ0(x ; ai , bi ) := ai < x < bi ;

ϕ1(x ; εj , εj+1) := (∃y ∈ Z) [εj+1 < |x − y | < εj ] .�




Inp-minimality vs. o-minimality

Lemma


But it turns out that unlike in the o-minimal case, it is possible tohave:

1 Closed definable sets with infinitely many connectedcomponents; or

2 Definable sets that are dense and codense in an interval.





Lemma









Lemma








An inp-minimal densely ordered group with a dense,codense predicate

Proposition

(Chatzidakis, Pillay) Let T = Th(R;<,+) and P be a unarypredicate. Then T has a model companion T ′ in the language{<,+,P} which is inp-minimal, and P is dense and codense.

Question

Is there a dense, codense P ⊆ R such that Th(R;<,+,P) isdp-minimal?

Note that if the answer is “no,” then an expansion of

Th(R;<,+) is o-minimal if and only if it is inp-minimal.





Proposition


Question








Proposition


Question







Unary definable functions

Next, consider a definable function f : M → M in an inp-minimaldensely ordered group.






Lemma

graph(f ) ⊆ M ×M has at most finitely many isolated points.






Lemma


Theorem

f is a union of finitely many continuous (partial) definablefunctions.






Lemma


Theorem


a

= f_1

= f_2





Lemma


Theorem


a

= f_1

= f_2

Idea of proof: define “limit functions” f1, f2, . . . of f , and showthat every fi is piecewise continuous.




The monotonicity theorem

Theorem

(Macpherson-Marker-Steinhorn, Arefiev) Let (M;<, . . .) be aweakly o-minimal structure and f : M → M a definable unaryfunction. Then there is a partition M = X0 ∪ . . . ∪ Xn−1 of M intodefinable convex subsets such that for each i < n, f � Xi iscontinuous and locally monotonic.

Theorem

(G.) Let (M;<,+, . . .) be an inp-minimal densely ordered groupand f : M → M a definable unary function. Then there is apartition M = X0 ∪ . . . ∪ Xn−1 of M into definable subsets suchthat for each i < n, f � Xi is continuous and locally monotonic.




The monotonicity theorem

Theorem

(Macpherson-Marker-Steinhorn, Arefiev) Let (M;<, . . .) be aweakly o-minimal structure and f : M → M a definable unaryfunction. Then there is a partition M = X0 ∪ . . . ∪ Xn−1 of M intodefinable convex subsets such that for each i < n, f � Xi iscontinuous and locally monotonic.

Theorem

(G.) Let (M;<,+, . . .) be an inp-minimal densely ordered groupand f : M → M a definable unary function. Then there is apartition M = X0 ∪ . . . ∪ Xn−1 of M into definable subsets suchthat for each i < n, f � Xi is continuous and locally monotonic.




Cell decomposition?

Question

Is there a nice cell decompostion theorem for inp-minimal DOGs?What is the right notion of a “cell” in these structures?

At least “cells” where one boundary is given by a continuousfunction seem OK:

Theorem

(G.) If B ⊆ Mn is an open box, and1 f , g : B → M are definable,

2 ∀x ∈ B [f (x) < g(x)], and

3 f is continuous,then {(x , y) : x ∈ B and f (x) < y < g(x)} has nonempty interior.




Cell decomposition?

Question



Theorem


2 ∀x ∈ B [f (x) < g(x)], and





Cell decomposition?

Question



Theorem


2 ∀x ∈ B [f (x) < g(x)], and





Dimension

Definition

If X ⊆ Mn is definable, dim(X ) is the largest r ∈ N such thatthere is a coordinate projection π : Mn → M r so that thetopological closure of π(X ) has nonempty interior in M r .




Dimension

Definition


This is the same as the usual definition of dimension in a (weakly)o-minimal theory, except that we take the closure of π(X ).




Dimension

Definition


This is the same as the usual definition of dimension in a (weakly)o-minimal theory, except that we take the closure of π(X ).

Example

If P ⊆ M is a definable dense subset of M, then dim(P) = 1.




Dimension

Definition


Example

If P ⊆ M is a definable dense subset of M, then dim(P) = 1.

Question

If X ⊆ Mn and f : X → Mm is any definable injective map, then isdim(X ) = dim(f (X ))?




No Dense Graphs

Definition

An ordered structure (M;<, . . .) has no dense graphs (or NDG) iffor every definable function f : Mn → M, graph(f ) ⊆ Mn+1 isnowhere dense.

Theorem

(G.) Any inp-minimal DOG has NDG.

So at least if f : Mn → Mm is definable and acts as the

identity on all but one coordinate, then f cannot collapse“dim.”




No Dense Graphs

Definition


Theorem







No Dense Graphs

Definition


Theorem







Definable functions on Mn: a conjecture

The following would help a lot in studying dimension, and possiblyin obtaining cell decompositions:

Conjecture

If f : Mn → M is a definable function in an inp-minimal DOG,then f is a finite union of (partial) continuous functions.




Definable functions on Mn: a conjecture

The following would help a lot in studying dimension, and possiblyin obtaining cell decompositions:

Conjecture

If f : Mn → M is a definable function in an inp-minimal DOG,then f is a finite union of (partial) continuous functions.




Dimension and burden

Question

In an inp-minimal DOG, is dim(X ) = bdn(X )?

Question

Does burden satisfy Lascar (in-)equalities in inp-minimal theories?What about in inp-minimal DOGs?




Dimension and burden

Question

In an inp-minimal DOG, is dim(X ) = bdn(X )?

Question

Does burden satisfy Lascar (in-)equalities in inp-minimal theories?What about in inp-minimal DOGs?




Works cited

Hans Adler, “Strong theories, burden, and weight,” preprint.2007-2008, available on author’s website.

Roman Arefiev, “On monotonicity for weakly o-minimalstructures,” preprint.

John Goodrick, “A monotonicity theorem for dp-minimaldensely ordered groups,” preprint.

Dugald Macpherson, David Marker, and Charles Steinhorn,“Weakly o-minimal structures and real closed fields,” Trans. ofthe Am. Math. Soc., vol. 352 (2000), pp. 5435–5483.

Alf Onshuus and Alexander Usvyatsov, “On dp-minimal stabletheories,” preprint, available on second author’s website.

Saharon Shelah, “Strongly dependent theories,” preprint,2007, arXiv:math/0504197v3.



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Monotonicity and dimension in dpinp-minimal groups · NIP and inp-minimalityUnary setsUnary functionsHigher dimensions Monotonicity and dimension in dpinp-minimal groups John Goodrick