Quelques tentatives de définir une notion générale de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques

2013

Abstract. According to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.In model theory, much attention has been drawn on algebraic structures which lie on the classifiable part of the dividing line provided by Shelah's stability theory, following the criterion: the models of some fixed uncountable cardinality of a theory T should not be classifiable if there are too many of them, such as if T is unstable, or stable non-superstable. Concerning theories with few countable models, less is known, even for N 0 -categorical groups (who have only one countable model up to isomorphism). All the less for the so called small structures who include all possible theories having fewer than continuum many countable models. Known results about small structures concern type-definable equivalence relations [15,8,9] Weakly small structures are introduced by Belegradek in [24] to provide a broad framework for both small and minimal structures: they include omega-stable structures but also Ko-categorical, minimal, and af-minimal ones recently introduced by Poizat in [19].In [24], Wagner shows that infinite small fields are algebraically closed, making the first successful use of the Cantor-Bendixson rank (the very weak analogue of Morley rank) in an algebraic context. He asks whether infinite weakly small fields are algebraically closed [24, Problem 12.5]. Following ideas of [24], an exploration of weakly small groups begins in [13] where it is noticed that a weakly small group G inherits locally several properties that omega-stable groups share globally. For instance G satisfies local descending chain conditions. Every definable subset of

Section: Weakly Small Fieldsmentioning

confidence: 85%

“…Wagner drew the same conclusion for a small field, as well as for a minimal field of positive characteristic [25,26]. Poizat extended the latter to d-minimal fields of positive characteristic [21]. Whether the same result holds even for a minimal field of characteristic zero is still unknown.…”

mentioning

confidence: 85%

Fields with few types

2013

“…A non minimal, d-minimal group. Recall that a minimal group is abelian [20,Reineke], and a d-minimal group is abelian-by-finite [17,Poizat]. Let M be a minimal group, and F a finite group of order d. Any semi-direct product M ⋊ F with a predicate interpreting M will do.…”

Section: Definition 2 (Belegradek)mentioning

confidence: 99%

“…G is an infinite group with bounded exponent and only one non-trivial conjugacy class. Such a group does not exist [20,17,Reineke]. For instance, as a group of exponent 2 is abelian, the group should have exponent a prime p = 2.…”

Section: (9) Final Contradictionmentioning

confidence: 99%

On properties of (weakly) small groups

2012

“…Il s'avère que les structures algébriques menues et les structures stables ont de nombreux caractères sinon semblables, du moins analogues. Les corps infinis tout d'abord ; qu'ils soient stables ou menus, ils n'ont pas de sous-groupes définissables d'indice fini, ni additifs, ni multiplicatifs (A. Macintyre [11] pour les corps stables, F. Wagner [17,Proposition 6] pour les corps menus et [9] pour les corps minces). Les corps menus infinis sont algébriquement clos (F. Wagner [17]), tandis que les corps superstables infinis sont algébriquement clos (G. Cherlin et S. Shelah [3]), que les corps stables infinis n'ont pas d'extensions de type Artin-Schreier (T. Scanlon [5,14]), et on conjecture qu'ils n'ont pas d'extensions séparables (voir [18]).…”

unclassified

Groupes Fins

2014

We investigate some common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left( {dc{l^{eq}}\left( A \right)} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left( {x,y} \right)$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left( A \right)$ofAof definable subgroups ofGover$acl\left( A \right)$which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions.