On The Classification of Geometries of Strongly Minim. Minimal Structures

Size: px

Start display at page:

Download "On The Classification of Geometries of Strongly Minim. Minimal Structures"

Dorothy Preston
5 years ago
Views:

1 On The Classification of Geometries of Strongly Minimal Structures BPGMT 2013

2 Definition - Strongly Minimal In this talk, when we say a set is definable in a model M, we mean it is definable in the language L M. Definition - Let M be a model. We say an infinite definable set D M is minimal if for every definable subset C D either C is finite or D \ C is finite. We say a theory T is strongly minimal if any model M = T is minimal as a definable subset of itself. We say a model is strongly minimal if its theory is.

3 Examples of Strongly Minimal Theories Examples: Infinite sets T h(z, s) Vector spaces over a fixed field F - In the language of +,0 and {f α } α F where f α is scalar multiplication by α. Affine spaces over a fixed field F - In the language {τ α } α F where τ α (a, b, c) = a + α(b c). ACF p

4 Pregeometries and Geometries Reminder: Let M be a model in a language L. Let A be some subset of M. We say a M is algebraic over A if there is some L A formula ϕ(x) such that ϕ(m) is finite and M = ϕ(a). The algebraic closure of A, is the set of algebraic elements over A. We denote it acl(a). Examples of acl: - In vector spaces acl(a) is simply span(a). - In ACF p, the acl(a) is the algebraic closure of the field generated by A.

5 Pregeometries and Geometries Definition - A Pregeometry (X, Cl) is a set X with an operator Cl : P (X) P (X) such that: 1. A Cl(A) 2. A B = Cl(A) Cl(B) 3. Cl(Cl(A)) = Cl(A) 4. Cl(A) = {Cl(A 0 ) A 0 A, A 0 is finite} (Finite Character) 5. a Cl(A, b) \ Cl(A) = b Cl(A, a) (Exchange Principal) For a model M, (M, acl) always satisfies properties 1-4. If M is also strongly minimal, then (M, acl) is a pregeometry.

6 Pregeometries and Geometries Let (X, Cl) be a pregeometry. Definition - We say A X is independent if a / Cl(A \ {a}) for any a A. Definition - We say A B is a basis for B closed, if Cl(A) = B and A is independent. Fact - Any two bases of a closed set B have the same cardinality. (this is proved using the exchange principal, exactly as in linear algebra) Definition - We define dim(b), the dimension of B, to be the cardinality of a basis of Cl(B). Definition - We define the localization of (X, Cl) at D to be (X, Cl D ) where Cl D (A) = Cl(A D). This is also a pregeometry. In the context of the acl pregeometry, localizing at D is achieved by simply adding D to the language as constants

7 Pregeometries and Geometries Definition - We say a pregeometry (X, Cl) is a Geometry if in addition Cl( ) = and Cl({x}) = x for any x X. A pregeometry (X, Cl) has a natural associated geometry: Let X 0 = X \ Cl( ) and say x y if x Cl({y}) (this is an equivalence relation by exchange). Consider X = X 0 / and Cl : X X with ( X, Cl) is a geometry. Cl(A) = {b/ b Cl({a a/ A}) }

8 Pregeometries and Geometries We distinguish several properties of pregeometries that are used for their classification: We say (X, Cl) is trivial if Cl(A) = a A Cl({a}) We say (X, Cl) is modular if for any finite dimensional sets A, B X dim(a B) = dim(a) + dim(b) dim(a B) We say (X, Cl) is locally modular if there is some a X such that (X, Cl {a} ) is modular. A vector space with the span operator is a great example for a modular pregeometry (modularity is exactly the Dimension Theorem from linear algebra). It is not hard to prove that a trivial pregeometry is modular. Try doing this by manipulating bases as you would in vector spaces.

9 Examples of acl Pregeometries Infinite sets - acl(a) = A so this theory has a trivial pregeometry T h(z, s) - acl({a}) is the connected component of a and acl(a) is the union of all connected components that intersect with A, so this theory has a trivial pregeometry Vector spaces over a fixed field F - acl(a) = span(a) and so these theories have non-trivial modular pregeometries Affine spaces over a fixed field F - If we localize at some point a (add a as a constant to the language) and declare it to be zero, then we get a vector space structure. So the pregeometries of these theories are non-trivial and locally modular. ACF p - The pregeometries of these theories are not locally modular. These are the source of all naturally-occurring examples of non-locally-modular geometries that we know of.

10 Zil ber s Conjecture and Its Refutation It has been thought that geometrical complexity must entail algebraic structure. In particular: Zil ber s Trichotomy Conjecture (1970 s) - The geometry of a strongly minimal structure M falls into one of the following three categories: 1 Trivial - M has no algebraic structure. (Combinatorial type) 2 Locally Modular (non-trivial) - there is an infinite vector space interpretable in M. (Linear type) 3 Rich (non-locally-modular) - M is essentially an algebraically closed field This conjecture has been (and still is) a central motivation and driving force in stability theory. It suggests an elegant hierarchical classification of strongly minimal geometries. Unfortunately, the conjecture was proven false.

11 Zil ber s Conjecture and Its Refutation Hrushovski, in his article A New Strongly Minimal Set, has refuted the Trichotomy conjecture. Hrushovski provides a robust mechanism for constructing strongly minimal structures with a non-locally-modular geometry, that do not interpret an infinite group. Because the geometry of such a structure is not locally-modular, it is not of the combinatorial or linear type. On the other hand, since the structure does not interpret a group (or even a semi-group), it introduces no algebraic structure. To clarify - Geometrically speaking, Hrushovski s construction fits between the linear type and the rich case (2 1 2 ) and algebraically, it fits between the trivial and linear cases (1 1 2 ). So geometrical and algebraic complexity are not necessarily in accordance. But perhaps we can fix the conjecture somehow so we still have a classification of all geometries of strongly minimal structures?

12 Zil ber s Conjecture and Its Refutation All structures constructed using Hrushovski s method share a common trait called CM-triviality. Informally, CM-triviality forbids the existence of a rich algebraic structure, like an infinite field. Question: Does the geometry of any strongly minimal structure M fall into one of these four categories? 1 Trivial - M has no algebraic structure. 2 Locally Modular (non-trivial) - there is an infinite vector space interpretable in M. 3 CM-trivial (non-locally-modular) - M has no algebraic structure 4 Rich (non-locally-modular or CM-trivial) - M is essentially an algebraically closed field Answer: No. (Hrushovski)

13 Hrushovski Fusion Hrushovski showed that any two strongly minimal theories (bar minimal technical requirements) can be fused into a single strongly minimal theory extending both initial theories. Theorem - Let T 1,T 2 be strongly minimal theories with DMP in disjoint countable languages L 1,L 2. Then there exists a strongly minimal theory T in L 1 L 2 such that T L i = T i. [Hrushovski 1992] In fact, countably many theories may be fused this way.

14 Difficulty of Reformulating the Trichotomy Conjecture The theorem leads to some unexpected/undesired results: There is no maximal strongly-minimal theory (at least not with DMP). For p q, there is a structure M = M, +,,, such that M, +, = ACF p and M,, = ACF q. So the new classification we suggested is still not enough. A structure that has algebraically closed fields of two distinct characteristics as reducts doesn t fit anywhere! In fact, Hrushovski s constructions are so robust and efficient at generating counter-examples, that 20 years later there is still no viable reformulation of the Trichotomy Conjecture.

15 Strongly Minimal Building Blocks So now that the introduction is over - what is it that I do? Hrushovski Fusions are the only current known obstruction to the classification of strongly minimal geometries. This makes one wonder whether there are certain minimal strongly-minimal theories that, in a way, are the building blocks of all strongly minimal theories. Perhaps these minimal theories can be classified. A good indication that a theory T is one of these basic building blocks is that any proper reduct of M = T has a trivial geometry. This notion is encouraged by naturally occurring examples like the theories of vector spaces over prime fields and the theories of algebraically closed fields.

16 Strongly Minimal Building Blocks Let M be Hrushovski s original construction presented in A New Strongly Minimal Set. A. Hasson (my advisor) and myself have examined a specific reduct M s of M. We have been able to show that M s is a proper reduct of M and that its geometry is not trivial. We expect that the geometry of M s will be identical to that of M, and that we will be able to repeat the process indefinitely. Thus, we will have an infinite chain of proper reducts with the same geometry as Hrushovski s original construction. As this does not eliminate the possibility of M having minimal non-trivial reducts, this does make it seem less likely. We hope that this will at least shed some light on where to find these minimal theories and whether they can be found at all.

17 Questions? Thank you!

An exposition of Hrushovski s New Strongly Minimal Set

An exposition of Hrushovski s New Strongly Minimal Set Martin Ziegler Barcelona, July 2011 In [5] E. Hrushovski proved the following theorem: Theorem 0.1 (Hrushovski s New Strongly Minimal Set). There