Author Archives: Artem Chernikov

Éric Jaligot, 1972 – 2013

Eric Jaligot

Page à la mémoire d’Eric Jaligot

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Model theoretic stability and definability of types, after A. Grothendieck

It had happened more than once that combinatorial model theoretic dividing lines introduced by Shelah were invented independently in different fields of mathematics. This curious note by Itai Ben Yaacov gives another example of this phenomenon:

We point out how the “Fundamental Theorem of Stability Theory”, namely the equivalence between the “non order property” and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck’s “Critères de compacité” from 1952. The familiar forms for the defining formulae then follow using Mazur’s Lemma regarding weak convergence in Banach spaces.

In a meeting in Kolkata in January 2013, the author asked the audience who had first defined the notion
of a stable formula and when, and to the expected answer replied that, no, it had been Grothendieck, in
the fifties.

MALOA “final” conference in Luminy

I had just attended the final MALOA conference “Logic and interactions“.  MALOA was a European network, which is now essentially finished (though the “final” meeting was not the last one, there will be a workshop in Manchester soon). The meeting took place at CIRM in Luminy, a wonderful place not only for mathematical reasons:

Luminy(you can see some more photos here).

As MALOA was about logic, rather than model theory, the topics of the talks were quite diverse, ranging from purely algebraic model theory (definable valuations and height bound in arithmetic nullstellensatz) to proof theory and even “logical description for behaviour analysis in aerospace systems”. Not sure how productive this diversity is, but at least it is entertaining.

Some talks that I found of particular interest are:

  • Talks by Deirdre Haskell  and Chris Laskowski on NIP, VC-density and connections to probability and combinatorics (in general one could safely add to this list some more subjects including set theory). These all are quite fascinating topics which deserve some postings in the future. There are already some examples of importing ideas from combinatorics (e.g. the beautiful (p,q)-theorem of Alon, Kleitman and Matousek) to prove model-theoretic results (e.g. UDTFS for NIP theories), but I believe that many more connections remain to be discovered.
  • Todor Tsankov spoke about generalizations of de Finetti’s theorem. Classical de Finetti’s theorem from probability theory says that a sequence of random variables is exchangeable if and only if it is independent and identically distributed over its tail sigma-algebra. Various multi-dimensional generalizations of this characterization form the so-called exchangeability theory. This theorem can viewed as providing a classification of all probability measures on 2^{\mathbb{N}} invariant under the action of S_{\infty}. Now, in a general situation, given a permutation group G acting on a countable set M, one can’t really hope to give any kind of “classification” of G-invariant measures on 2^{\mathbb{N}} as we are in the context of the general ergodic theory. However, it appears that if the group G is sufficiently large compared to the index set, one can arrive at stronger results. Todor’s approach is to consider oligomorphic groups, i.e. such that the action of G on M^n has only finitely many orbits for each n. These groups are familiar to model theorists as automorphism groups of \omega-categorical structures. Todor provides a classification in the case when the underlying structure has trivial algebraic closure, and gives some promising partial results in the general case. In fact, this subject appears to have a lot to do with model theory. I am involved in a project together with Itai Ben Yaacov, of an abstract model theoretic approach to de Finetti’s theory in terms of the forking calculus, canonical bases and Morley sequences in the context of an arbitrary stable first-order theory, in the sense of continuous logic (which specializes to the classical case considering the theory of [0, 1]-valued random variables equipped with the L^1 metric).

Also I gave what was probably my last talk as a “student”. I spoke about some new results with Pierre Simon and Anand Pillay concerning definable topological dynamics in NIP theories. The slides are available here. We show that notions like definable (extreme) amenability of a definable group, as well as various model theoretic components, are not affected by adding externally definable sets to the picture (that is, passing to a Shelah’s expansion of a model). These facts appear to have some applications to the questions of Newelski on describing G/G^{00} in terms of the so-called Ellis group.

“Model theory and applications to geometry”, Satellite workshop associated with LC 2013, Lisbon, 18th-19th of July 2013

The aim of the workshop is to give the opportunity to the people attending Logic Colloquium 2013 and to all the people interested, to hear more about the recent developments in model theory and its applications and connections to real geometry.

List of confirmed speakers:

Raf Cluckers (Université de Lille and KULeuven)
Georges Comte (Université de Savoie – Chambéry)
Antongiulio Fornasiero (Seconda Università di Napoli)
Isaac Goldbring (UIC)
François Loeser (Université Pierre et Marie Curie)
Chris Miller (Ohio State University)
Giovanni Morando (Università di Padova)
Jean-Philippe Rolin (Université de Bourgogne)

You can find information about travelling and accommodation on the webpage of the workshop: http://ptmat.fc.ul.pt/~tservi/Workshop%202013/

REGISTRATION: if you are planning on attending this event, please contact tamara.servi@gmail.com

We hope to see you in Lisbon!

The Model Theory Group at CMAF
Universidade de Lisboa

Postdoc in model theory (Caserta, Italy)

I think that we might use this platform to spread information about upcoming meetings and position openings. The following is a message from Paola D’Aquino (Paola.DAQUINO@unina2.it)

Dear All,
there is a two year postdoc position in model theory in my department in Caserta. It will be advertised by end of April and the deadline for applying is around end of May. It’s not required to have a Ph.D. by the
deadline for application. The salary is 25,000 per year before taxes. If you know of anyone who can be interested they can write to me for more information.
Best wishes,
Paola

Some counterexamples for forking, dividing, invariance

At the end of my paper with Itay Kaplan “Forking and dividing in NTP2 theories” we had asked several questions, admittedly without giving them much thought. Since 2008 when the paper went in circulation, some people had actually shown interest in those questions. By now two of them are known to have negative answers, one due to Gabriel Conant and one by myself, with very easy examples. I’d like to have them written down for a reference somewhere, so I’ve thought this might be an appropriate place.

good forking

Question 1. (rephrased as more elaborate latex is not available here): Is it true that in a simple theory, every type has a global Lascar-invariant extension.

I recall that a complete global type {p\left(x\right)\in S\left(\mathbb{M}\right)} is Lascar-invariant over a small set {A} if whenever {\phi\left(x,a\right)\in p} and {b} has the same Lascar strong type over {A} as {a}, then {\phi\left(x,b\right)\in p}. Having the same Lascar strong type means that {a} is equivalent to {b} with respect to every equivalence relation with boundedly many classes which is {\mbox{Aut}\left(\mathbb{M}/A\right)}-invariant.

This property is true in the random graph, for example – any type can be extended to a global one without adding any new edges. This is also true in any extensible NIP theory, say in any stalbe theory, any {o}-minimal theory (e.g. real closed fields) or any {C}-minimal theory (e.g. algebraically closed valued fields), as well as in any theory with definable Skolem functions (e.g. {p}-adics). A theory is extensible if every type does not fork over its domain. However, the crucial point is that this property need not be preserved in reducts of the theory, which immediately gives an easy simple counterexample.

Example. Let {T} be the reduct of the random graph given by the ternary relation {R(x,y,z)} which holds if and only if {x\neq y\neq z} and the number of edges between vertices in the set {\left\{ x,y,z\right\} } is odd.

Claim.

  1. {T} is supersimple of {SU}-rank {1}. Thus, Lascar strong type is determined by the strong type.
  2. For any set {A}, {\mbox{acl}(A)=A}.
  3. All pairs of different elements have the same type over {\emptyset}.
  4. Let {M\models T} and {p\in S(M)}. Then {p} is not Lascar-invariant over {\emptyset}.

Proof: (1) is because {T} is definable on the set of singletons in the random graph, which is supersimple of {SU}-rank 1. Now it is a well-known fact that Lascar strong type is determined by the strong type in supersimple theories.

(2) is easy to see, and (3) is by back-and-forth.

(4) Assume {p} is Lascar-invariant over {\emptyset}, thus invariant over it by (1) and (2). Let {a\models p}, then by (3) either {\models R(a,b,c)} for all {b\neq c\in M} or {\models\neg R(a,b,c)} for all {b\ne c\in M}. In the first case, let {b\neq c\neq d\in M} satisfy {\neg R(b,c,d)}. Then it is easy to see that {\not\models R(a,b,c)\land R(a,c,d)\land R(a,b,d)} — a contradiction. In the other case, take {b,c,d} satisfying {R(b,c,d)} and check that {\not\models\neg R(a,b,c)\land\neg R(a,c,d)\land\neg R(a,b,d)} — a contradiction again. \Box

Thus, by (4) the unique type over the empty set has no global Lascar-invariant extension.

There are various modifications of the question which still make sense, and also one can ask if this property holds in particular algebraic structures of interest. I have some things to say about it, but not this time.

Question 2. “Can similar results be proved for NSOP theories?”

Here “similar results” refers to the main result of the paper, that is that in an NTP2 theory a formula divides over an extension base if and only if it forks over it. Now, Gabe shows in “Forking and dividing in Henson graphs” that it is not the case for the triangle-free random graph. From my own experience, triangle-free random graph seems to demonstrate the failure of all the phenomena which holds for NTP2 theories.

Example. Let {T} be the theory of the triangle-free random graph, and let {b_{0}\neq b_{1}\neq b_{2}\neq b_{3}}. Let {\phi\left(x,b_{0}b_{1}b_{2}b_{3}\right)=\bigvee_{i<j<4}\left(xRb_{i}\land xRb_{j}\right)}.

Claim.

  1. {xRb_{i}\land xRb_{j}} divides over {\emptyset} for any {i<j}.
  2. {\phi\left(x,b_{0}b_{1}b_{2}b_{3}\right)} does not divide over {\emptyset}.
  3. {\emptyset} is an extension base.
  4. {T} is {\mbox{SOP}_{3}} but {\mbox{NSOP}_{4}}.
  5. However, forking and dividing are the same for complete types.

See Gabe’s article for details and for the general case of Henson graphs.

Still the following part of the question remains open:

Problem.

  1. Is forking=dividing for complete types?
  2. Is forking equal to dividing for formulas in NTP1 over models?

On the SOPn hierarchy inside NTP2

In this post I would like to popularize a certain question around Shelah’s classification theory of unstable (and even non-simple) theories.

tree property

1. Tree properties of the first and second kind, {\mbox{NTP}_{1}} and {\mbox{NTP}_{2}}

As usual we are in a very big and saturated monster model {\mathbb{M}} of a complete first-order theory {T}.

Definition. A formula {\phi(x,y)} has {\mbox{TP}_{1}} if there is a tree {\left(a_{\eta}\right)_{\eta\in\omega^{<\omega}}} of tuples and {k\in\omega} such that:
{\left\{ \phi(x,a_{\eta|i})\right\} _{i\in\omega}} is consistent for any {\eta\in\omega^{\omega}}
{\left\{ \phi(x,a_{\eta_{i}})\right\} _{i<k}} is inconsistent for any mutually incomparable {\eta_{0},...,\eta_{k-1}\in\omega^{<\omega}}.
Otherwise we say that {\phi(x,y)} is {\mbox{NTP}_{1}}, and {T} is {\mbox{NTP}_{1}} if every formula is.

Definition. A formula {\phi(x,y)} has {\mbox{TP}_{2}} if there is an array {\left(a_{\alpha,i}\right)_{\alpha,i<\omega}} of tuples such that {\left\{ \phi(x,a_{\alpha,i})\right\} _{i<\omega}} is {2}-inconsistent for every {\alpha<\omega} and {\left\{ \phi(x,a_{\alpha,f(\alpha)})\right\} _{\alpha<\omega}} is consistent for any {f:\,\omega\rightarrow\omega}. Otherwise we say that {\phi(x,y)} is {\mbox{NTP}_{2}}, and {T} is {\mbox{NTP}_{2}} if every formula is.

It is an easy exercise to see that each of this properties implies the usual tree property, that is the failure of simplicity. An important insight of Shelah is that failure of simplicity always occurs in one of these two explicit ways.

Theorem. {T} is simple if and only if it is both {\mbox{NTP}_{1}} and {\mbox{NTP}_{2}}.

These tree properties were introduced and studied by Shelah in [1] and [2]. {\mbox{NTP}_{2}} re-appeared on the scene after it arose in the work on forking and dividing in NIP theories in [3] and was studied further in [4] and [5]. {\mbox{NTP}_{1}} re-appears in [6]. I will probably return to these in future postings.

2. On the {\mbox{NSOP}_{n}} hierarchy inside {\mbox{NTP}_{2}}

We recall the definition of {\mbox{SOP}_{n}} for {n\geq2} from [7,Definition 2.5], another hierarchy introduced by Shelah in order to study non-simple theories without the strict order property:

Definition. 1. Let {n\geq3}. A formula {\phi\left(x,y\right)} has {\mbox{SOP}_{n}} if there are tuples {\left(a_{i}\right)_{i\in\omega}} such that:

  • There is an infinite chain: {\models\phi\left(a_{i},a_{j}\right)} for all {i<j<\omega}
  • There are no cycles of length {n}: {\models\neg\exists x_{0}\ldots x_{n-1}\bigwedge_{j=i+1\left(\mod n\right)}\phi\left(x_{i},x_{j}\right)}.

2. {\phi\left(x,y\right)} has {\mbox{SOP}_{2}} if and only if it has {\mbox{TP}_{1}}.
3. {\mbox{SOP}\Rightarrow \mbox{SOP}_{n+1}\Rightarrow \mbox{SOP}_{n}\Rightarrow\ldots\Rightarrow\mbox{SOP}_{3}\Rightarrow\mbox{SOP}_{2}\Rightarrow \mbox{not simple}}.
4. By Shelah’s theorem we see that restricting to {\mbox{NTP}_{2}} theories, the last 2 items coincide.

The following are standard examples showing that the {\mbox{SOP}_{n}} hierarchy is strict for {n\geq3}:

Example. [7, Claim 2.8]
1. The usual example of a theory which is not-simple and {\mbox{NSOP}_{3}} is the model companion of the theory of a parametrized family of equivalence relations with infinitely many infinite classes (see [8]).
2. For {n\geq3}, let {T_{n}} be the model completion of the theory of directed graphs (no self-loops or multiple edges) with no directed cycles of length {\leq n}. Then it has {\mbox{SOP}_{n}} but not {\mbox{SOP}_{n+1}}.
3. For odd {n\geq3}, the model completion of the theory of graphs with no odd cycles of length {\leq n}, has {\mbox{SOP}_{n}} but not {\mbox{SOP}_{n+1}}.
4. Consider the model companion of a theory in the language {\left(<_{n,l}\right)_{l\leq n}} saying:

  • {x<_{n,m-1}y\rightarrow x<_{n,m}y},
  • {x<_{n,n}y},
  • {\neg\left(x<_{n,n-1}x\right)},
  • if {l+k+1=m\leq n} then {x<_{n,l}y\,\land\, y<_{n,k}z\,\rightarrow\, x<_{n,m}z}.

 It eliminates quantifiers.

However, all these examples have {\mbox{TP}_{2}}.
Proof:
(1) It is immediate that the formula {E\left(x;y,i\right)} has {\mbox{TP}_{2}}.
(2) Let {\phi\left(x,y_{1}y_{2}\right)=xRy_{1}\land y_{2}Rx}. For {i\in\omega} we choose sequencese {\left(a_{i,j}b_{i,j}\right)_{j\in\omega}} such that {\models R\left(a_{i,j},b_{i,k}\right)} and {R\left(b_{i,j},a_{i,k}\right)} for all {j<k\in\omega}, and these are the only edges around — it is possible as no directed cycles are created. Now for any {i}, if there is {c\models\phi\left(x,a_{i,0}b_{i,0}\right)\land\phi\left(x,a_{i,1}b_{i,1}\right)}, then we would have a directed cycle {c,b_{i,0},a_{i,1}} of length {3} — a contradiction. On the other hand, given {i_{0}<\ldots<i_{n}} and {j_{0},\ldots,j_{n}} there has to be an element {a\models\bigwedge_{\alpha\leq n}\phi\left(x,a_{i_{\alpha},j_{\alpha}}b_{i_{\alpha},j_{\alpha}}\right)} as there are no directed cycles created. Thus {\phi\left(x,y_{1}y_{2}\right)} has {\mbox{TP}_{2}}.
(3) and (4) Similar.

This naturally leads to the following question (which I originally asked in [4]):

Question: Is the {\mbox{SOP}_{n}} hierarchy strict for {\mbox{NTP}_{2}} theories? Note that the strictness of the implication {\mbox{SOP}_{3}\Rightarrow \mbox{SOP}_{2}} is open even in general.

Even an example of a non-simple theory with {\mbox{NSOP}} and {\mbox{NTP}_{2}} is missing. In [2, §7, Exercise 7.12] Shelah suggests an example of such a theory as an exercise. However, I couldn’t make sense of the suggested example. Perhaps someone else would?

References

  1. Saharon Shelah, “Simple unstable theories”, Ann. Math. Logic, http://dx.doi.org/10.1016/0003-4843(80)90009-1
  2. Saharon Shelah, “Classification theory and the number of nonisomorphic models”, North-Holland Publishing Co., 1990
  3. Artem Chernikov and Itay Kaplan, “Forking and dividing in NTP2 theories”, J. Symbolic Logic, 2012
  4. Artem Chernikov, “Theories without the tree property of the second kind”, http://arxiv.org/abs/1204.0832
  5. Itai Ben Yaacov and Artem Chernikov, “An independence theorem for NTP2 theories”, arXiv:1207.0289
  6. Byunghan Kim and Hyeung-Joon Kim, “Notions around tree property 1”, Annals of Pure and Applied Logic, 2011
  7. Saharon Shelah, “Toward classifying unstable theories”, Ann. Pure Appl. Logic, 1996
  8. Saharon Shelah and Alex Usvyatsov, “More on SOP1 and SOP2”, arXiv:math/0404178