E. Hrushovski, Simplicity and the Lascar group, preprint, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....all of the material in this paper is taken from the Ph. D. Thesis [Pa95] of the Second Author at the University of East Anglia. A preliminary version of the paper has been available since 1995. Since then, the construction of unstable structures in [Hr88] has acquired an additional signi cance. In [Hr97], Hrushovski pointed out that these constructions give rise to simple, unstable 0 categorical structures which are not modular. To ensure that the structures we produce in Theorems 1 and 2 are simple we require an extra condition on the function f used in the construction. If f(3t) f(t) 1 ....

....constructions give rise to simple, unstable 0 categorical structures which are not modular. To ensure that the structures we produce in Theorems 1 and 2 are simple we require an extra condition on the function f used in the construction. If f(3t) f(t) 1 (for t k) then the argument in [Hr97] shows that M is supersimple, of SU rank k in the case of the structures for Theorem 1, and SU rank 1 for those of Theorem 2 (more detail can be found in [Ev00, Theorem 3.6] It seems clear that our arguments can be adjusted to accommodate this extra condition. We conclude the Introduction with ....

Ehud Hrushovski. Simplicity and the Lascar group, Preprint, 1997.

....due to the lack of stationarity. What is known is Hrushovski s counterexample. The example shows Zilber s result on categorical strongly minimal structures can not be generalized in the context of categorical rank 1 structures. Namely, there is non 1 based rank 1 categorical structure [12]. In this paper we attempt to develop initial geometric simplicity theory using classical results in geometry as essential tools. We somehow succeed to gain some of fruitful positive results, and examples. In the paper, we mainly study the solution set D of non trivial 1 based (simple) SU rank 1 ....

....dim(abuv) dim(cl(ab) cl(uv) 3) This nishes the proof. In above lemma, the assumption that D is modular is crucial. We shall see that even local modularity of D is not sucient to supply the conclusion of the lemma (Section 2) There are other 1 based (3. 7) and non 1 based examples too [12] (where closure of some pair of points can be 2 or 3 set. By now, we have the following theorems due to Fact 1.5 and previous lemma. Theorem 1.9. If D is modular, then either D trivial, or the geometry of D( D 0 ) is projective over some division ring. Either case, D; cl) is homogeneous. ....

E. Hrushovski, `Simplicity and the Lascar group', preprint (1998).

....Lascar strong types (E L classes) took the place of strong types. Kim [4] subsequently showed that simple theories are G compact. The second author, in [6] defined a topology on Gal L (T ) in the case where T is G compact, making Gal L (T ) into a compact (Hausdorff) topological 2 group. In [2], Hrushovski gave another account of the topology, working directly with GalKP (whether T is G compact or not) In fact in that paper the EKP notation was introduced. Similar things were done in [7] The main point was that the spaces S=EKP or even X=EKP are naturally equipped with compact ....

....Hausdorff topologies (the closed sets being precisely the typedefinable sets) There has been considerable attention paid to the issue of proving that EKP = E Sh in certain situations. For example in [1] this is proved for supersimple theories. The simple case is still open although Hrushovski [2] found a counterexample in the more general (non first order) context of Robinson theories. The current paper is concerned with the issue of when and how E L differs from EKP , in particular the existence of non G compact theories. The starting point for our work was the discovery by the fourth ....

E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

....Lascar strong types (E L classes) took the place of strong types. Kim [4] subsequently showed that simple theories are G compact. 2 The second author, in [6] defined a topology on Gal L (T ) in the case where T is G compact, making Gal L (T ) into a compact (Hausdor#) topological group. In [2], Hrushovski gave another account of the topology, working directly with GalKP (whether T is G compact or not) In fact in that paper the EKP notation was introduced. Similar things were done in [7] The main point was that the spaces S EKP or even X EKP are naturally equipped with compact ....

....Hausdor# topologies (the closed sets being precisely the typedefinable sets) There has been considerable attention paid to the issue of proving that EKP = E Sh in certain situations. For example in [1] this is proved for supersimple theories. The simple case is still open although Hrushovski [2] found a counterexample in the more general (non first order) context of Robinson theories. The current paper is concerned with the issue of when and how E L di#ers from EKP , in particular the existence of non G compact theories. The starting point for our work was the discovery by the fourth ....

E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

....[4] where Lascar strong types (E L classes) took the place of strong types. Kim [3] subsequently showed that simple theories are G compact. Lascar in [5] defined a topology on Gal L (T ) in the case where T is G compact, making Gal L (T ) into a compact (Hausdorff) topological group. In [2], Hrushovski gave another account of the topology, working directly with GalKP (whether T is G compact or not) In fact in that paper the EKP notation was introduced. Similar things were done in [6] The main point was that the spaces S=EKP or even X=EKP are naturally equipped with compact ....

....Hausdorff topologies (the closed sets are precisely the typedefinable sets) There has been considerable attention paid to the issue of proving that EKP = E Sh in certain situations. For example in [1] this is proved for supersimple theories. The simple case is still open although Hrushovski [2] found a counterexample in the more general (non first order) context of Robinson theories. The current paper is concerned with the issue of when and how E L differs from EKP , in particular the existence of non G compact theories. The starting point was the discovery by the fourth author of such ....

E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

....for the category of existentially closed models of an arbitrary universal theory T , assuming a suitable notion of simplicity . Under the additional assumption that T has the amalgamation property (AP) and joint embedding property (JEP) Shelah did develop forking in some form. Hrushovski ([4]) rediscovered this latter class of theories (universal T with AP and JEP) calling them Robinson theories, and pointing out that all model theoretic methods should apply to the category of e.c. models of such T . For Robinson theories, quantifier free types are the main object of study, and as ....

E. Hrushovski, Simplicity and the Lascar group, preprint 1998.

....related results as well as the genesis of the current paper (which is a bit complicated) Kim [10] proved that in any countable theory with only countably many types, any type definable equivalence relation (over a finite set) is a conjunction of definable equivalence relations. Hrushovski ([6]) showed that if one was willing to work in suitable non first order contexts (so called Robinson structures) then there are supersimple theories which do not eliminate (even bounded) hyperimaginaries. Buechler [1] proved that in simple theories satisfying an additional technical property ....

E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

....for its support and hospitability 1 groups. In section 4 we study a certain group introduced in [5] which we call the Galois group of T , develop a Galois theory and make the connection with the ideas in section 3. In sections 3 and 4 there is some overlap with parts of Hrushovski s paper [2]. In section 5, we show that if T satisfies some mild assumptions, then for a suitably saturated model M of T , the subgroups of Aut( M) which are stabilizers of bounded closed quasifinitary hyperimaginaries can be recognized from the abstract group structure of Aut( M ) All these notions ....

E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

.... superstable group is abelian by finite [1] and has finite rank (in fact by [3] every categorical superstable theory is even one based of finite Morley rank) In the case of a supersimple categorical theory, much less is known except that things do not turn out that nicely: recently, Hrushovski [5] has constructed a simple categorical theory of SU rank 1 whose geometry is not locally modular. It is unknown whether a supersimple categorical theory can have infinite rank. The question therefore arises whether at least in the group case the results from the superstable case generalize. ....

Ehud Hrushovski, Simplicity and the Lascar group, notes (1997).

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E. Hrushovski, Simplicity and the Lascar group, preprint, 1998.

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E. Hrushovski, `Simplicity and the Lascar group', preprint (1998).

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E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

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E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

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E. Hrushovski, Simplicity and the Lascar group, preprint 1997.

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