In this paper, we shall survey results about the group-theoretic properties of stable groups. These can be classified into three main categories, according to the strength of the assumptions needed: chain conditions, generic types, and some form of rank. Each category has its typical application: Chain conditions often allow us to deduce global properties from local ones, generic properties are used to get definable groups from undefinable ones, and rank is necessary to interpret fields in certain group actions. While originally (...) the main input came from algebraic group theory, founded on the similarities between rank and dimension, increasingly methods borrowed from finite group theory have come to play a rôle, emphasizing the study of characteristic subgroups and characteristic families of subgroups. (shrink)

Let T be simple, work in Ceq over a boundedly closed set. Let p ∈ S(θ) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose (p, Q) is ACL-embedded in Q (see definitions below). Then Aut(p/Q) with its action on pC is type-definable in Ceq over θ. In particular, if p ∈ S(θ) is internal in a stably-embedded type-definable set Q, and pC υ Q is stably-embedded, then Aut(p/Q) is type-definable with its action (...) on pC. (shrink)

In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence ( $a_{i}|i \leq n^{*}$ ) $\subseteq dcl^{eq}$ (a) with $a_{n}*$ = a such that for every $i \leq n*$ , if $p_{i} = tp(a_{i}/{a_{i}|j < i}$ ) then $Aut(p_{i}/Q)$ is type-definable with its action on $p_{i}^{c}$ . A unidimensional simple theory eliminates the quantifier $\exists^{\infty}$ and either interprets (in $C^{eq}$ ) an infinite type-definable group (...) or has the property that ACL(Q) = C for every infinite definable set Q. (shrink)

Nous isolons des propriétés valables dans certaines théories de purs corps ou de corps munis d’opérateurs afin de montrer qu’une théorie est simple lorsque les clôtures définissables et algébriques sont contrôlées par une théorie stable associée.

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.

The binding group theorem for stable theories is partially extended to the simple context. Some results concerning internality are proved. We also introduce a ‘small’ normal subgroup G0+ of the automorphism group and show that if p is Q-internal then it has a finite exponent and G/G0+ is interpretable.

First, it is pointed out how the author's new differential Galois theory contributes to the understanding of the differential closure of an arbitrary differential field . Secondly, it is shown that a superstable differential field has no proper differential Galois extensions.

T is stable. We define the notion of meager regular type and prove that a meager regular type is locally modular. Assuming I < 2o and G is a definable abelian group with locally modular regular generics, we prove a counterpart of Saffe's conjecture. Using these results, for superstable T we prove the conjecture of vanishing multiplicities. Also, as a further application, in some additional cases we prove a conjecture regarding topological stability of pseudo-types over Q.

We investigate the interaction between the product of invariant types and domination–equivalence. We present a theory where the latter is not a congruence with respect to the former, provide sufficient conditions for it to be, and study the resulting quotient when it is.

We describe the ind- and pro- categories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.

Assume that G is a definable group in a stable structure M. Newelski showed that the semigroup SG of complete types concentrated on G is an inverse limit of the ∞-definable semigroups SG,Δ. He also showed that it is strongly π-regular: for every p∈SG,Δ, there exists n∈N such that pn is in a subgroup of SG,Δ. We show that SG,Δ is in fact an intersection of definable semigroups, so SG is an inverse limit of definable semigroups, and that the latter (...) property is enjoyed by all ∞-definable semigroups in stable structures. (shrink)

We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42, 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from (...) [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281, 2016]. (shrink)

Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the (...) failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split. (shrink)

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard—Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In addition, making (...) use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations. (shrink)

Assume T is unidimensional, 1-based and every minimal type in T is locally finite. If H is an Λ -definable irreducible group, we find an irreducible supergroup G of H in acleq such that any connected subgroup of Gn, n < ω, is the connected component of a subgroup linearly defined over the ring End*. In some cases we can take G = H.

We extend the dichotomy between 1-basedness and supersimplicity proved in Shami :309–332, 2011). The generalization we get is to arbitrary language, with no restrictions on the topology [we do not demand type-definabilty of the open set in the definition of essential 1-basedness from Shami :309–332, 2011)]. We conclude that every hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the (...) language is countable. (shrink)

Suppose T is a weakly minimal theory and p a strong 1-type having locally finite but nontrivial geometry. That is, for any M [boxvR] T and finite Fp, there is a finite Gp such that acl∩p = gεGacl∩pM; however, we cannot always choose G = F. Then there are formulas θ and E so that θεp and for any M[boxvR]T, E defines an equivalence relation with finite classes on θ/E definably inherits the structure of either a projective or affine space (...) over some finite field. We then specify what other structure θ/E may inherit: there is some collection of definable subspaces of finite codimension and some set of algebraic points, which in the affine case may be in the canonically associated vector space, Up to acl, no further structure is possible. If we assume T is weakly minimal and has a strong type p as above, and also that T is unidimensional, we obtain a global description of any model of T in terms of those structures mentioned in the previous paragraph. (shrink)

Let G be a stable abelian group with regular modular generic. We show that either 1. there is a definable nongeneric K ≤ G such that G/K has definable connected component and so strongly regular generics, or 2. distinct elements of the division ring yielding the dependence relation are represented by subgroups of G × G realizing distinct strong types (when regarded as elements of G eq ). In the latter case one can choose almost 0-definable subgroups representing the elements (...) of the division ring. We find a bound $((G: G^0))$ for the size of the division ring in case G has no definable subgroup K so that G/K is infinite with definable connected component. We show in case (2) that the group G/H, where H consists of all nongeneric points of G, inherits a weakly minimal group structure from G naturally, and Th(G/H) is independent of the particular model G as long as G/H is infinite. (shrink)

We generalize tools and results from first order stable theories to groups inside a simple stable strongly homogeneous model.

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD(A), where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an∈ P and finite subset C ⊆ Q, but (...) [Formula: see text] for some independent a1, …, an, an+1∈ P and some finite subset C ⊆ Q.Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] interprets a non-classical group, or n = 1,2,3 and•If n = 1 then G is abelian and acts regularly on P′.•If n = 2 the action of G on P′ is isomorphic to the affine action of K ⋊ K* on the algebraically closed field K.•If n = 3 the action of G on P′ is isomorphic to the action of PGL2(K) on the projective line ℙ1(K) of the algebraically closed field K.We prove a similar result for excellent classes. (shrink)

We prove that any countable simple unidimensional theory T is supersimple, under the additional assumptions that T eliminates hyperimaginaries and that the $D_\phi-ranks$ are finite and definable.