29 research outputs found
A remark on Cantor derivative
It is shown that, modulo an equivalence relation induced by finite
correspondences preserving Cantor rank, the class of topological spaces is an
integral semi-ring on which the Cantor derivative is precisely a derivation
On supersimple groups
International audienceAn infinite group with supersimple theory has a finite series of definable groups whose factors are infinite and either virtually-FC or virtually-simple modulo a finite FC-centre. A group which is type-definable in a supersimple theory has a finite series of relatively definable groups whose factors are either abelian or simple groups. In this decomposition, the non-abelian simple factors are unique up to isomorphism
ON THE RADICALS OF A GROUP THAT DOES NOT HAVE THE INDEPENDENCE PROPERTY
International audienceWe give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable and whose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013
, and division rings of prime characteristic
Combining a characterisation by BĂ©lair, Kaplan, Scanlon and Wagner of certain valued fields of characteristic with Dickson's construction of cyclic algebras, we provide examples of noncommutative division ring of characteristic and show that an division ring of characteristic has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite fields have no Artin-Schreier extension. The result extends to division rings of characteristic , using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern or simple difference fields
Fields and rings with few types
Let R be an associative ring with possible extra structure. R is said to be
weakly small if there are countably many 1-types over any finite subset of R.
It is locally P if the algebraic closure of any finite subset of R has property
P. It is shown here that a field extension of finite degree of a weakly small
field either is a finite field or has no Artin-Schreier extension. A weakly
small field of characteristic 2 is finite or algebraically closed. Every weakly
small division ring of positive characteristic is locally finite dimensional
over its centre. The Jacobson radical of a weakly small ring is locally
nilpotent. Every weakly small division ring is locally, modulo its Jacobson
radical, isomorphic to a product of finitely many matrix rings over division
rings
Propriétés algébriques des structures menues ou minces, rang de Cantor Bendixson, espaces topologiques généralisés
Abstract. Small structures appear in the '60s together with Vaught's conjecture. Weakly small structures include both minimal and small structures. Definable sets in a weakly small structure are ranked by Cantor-Bendixson rank. We show computational properties of this rank, which imply a local descending chain condition on acl(0)-definable subgroups, and introduce a notion of local almost stabiliser. We deduce algebraic properties of weakly small structures. Among them, a weakly small field of positive characteristic is locally finite dimensional over its centre, and an infinite weakly small group has an infinite abelian subgroup. We then turn to small type-definable structures, showing that finitary small type 0-de_nable groups are the intersection of definable groups. We extend the result to finitary small type 0- definable monoids, rings, fields, categories and groupoids. We give local definability results concerning groups and fields type definable over an arbitrary set of parameters in small and simple theories. Finally, we reintroduce the Cantor Bendixson rank in its topological context, and show that the Cantor derivative can be seen as a derivation in a semi-ring of topological spaces. In an attempt to find a global Cantor rank for stable structures, we try to eliminate the word denumerable, omnipresent when one does topology, by replacing it by a regular cardinal k. We develop the notions of k-metrisable space, k-topology, k-compactness etc. and show an analogue of Urysohn's metrisability lemma and Cantor-Bendixson theorem.Les structures menues apparaissent dans les années 60 en lien avec la conjecture de Vaught. Les structures minces englobent à la fois les structures minimales et menues. Les ensembles définissables d'une structure mince sont rangés par le rang de Cantor-Bendixson. Nous présentons des propriétés de calcul de ce rang, une condition de chaîne descendante locale sur les groupes acl(0)-définissables ainsi qu'une notion de presque stabilisateur local, et en déduisons des propriétés algébriques des structures minces : un corps mince de caractéristique positive est localement de dimension finie sur son centre, et un groupe mince infini a un sous groupe abélien infini. Nous nous intéressons ensuite aux structures menues infiniment définissables, et montrons que les groupes d'arité finie infiniment 0-définissable sont l'intersection de groupes définissables. Nous étendons le résultat aux demi-groupes, anneaux, corps, catégories et groupoïdes infiniment 0-définissables, et donnons des résultats de définissabilité locale pour les groupes et corps simples et menus, infiniment définissables sur des paramètres quelconques. Enfin, nous réintroduisons le rang de Cantor dans son contexte topologique et montrons que la dérivée de Cantor peut être vue comme un opérateur de dérivation dans un semi-anneau d'espaces topologiques. Dans l'idée de trouver un rang de Cantor global pour les théories stables, nous essayons de nous débarrasser du mot dénombrable omniprésent lorsque l'on fait de la topologie, en le remplaçant par un cardinal régulier k. Nous développons une notion d'espace k-métrique, de k-topologie, de k-compacité etc. et montrons un k-analogue du lemme de métrisabilité d'Urysohn, et du théorème de Cantor-Bendixson
On properties of (weakly) small groups
A group is small if it has countably many complete -types over the empty
set for each natural number n. More generally, a group is weakly small if
it has countably many complete 1-types over every finite subset of G. We show
here that in a weakly small group, subgroups which are definable with
parameters lying in a finitely generated algebraic closure satisfy the
descending chain conditions for their traces in any finitely generated
algebraic closure. An infinite weakly small group has an infinite abelian
subgroup, which may not be definable. A small nilpotent group is the central
product of a definable divisible group with a definable one of bounded
exponent. In a group with simple theory, any set of pairwise commuting elements
is contained in a definable finite-by-abelian subgroup. First corollary : a
weakly small group with simple theory has an infinite definable
finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal
solvable group A of derived length n is contained in an A-definable almost
solvable group of class n
On enveloping type-definable structures
International audienceWe observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a 0-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, rings, categories, groupoids and preorders which are 0-type-definable in a small structure. For an A-type-definable group G_A (where the set A may be infinite) in a small and simple structure, we deduce that (1) if G_A is included in some definable set X such that boundedly many translates of G_A cover X, then G_A is the conjunction of definable groups. (2) for any finite tuple g in G_A, there is a definable group containing g
Small skew fields
International audienceWedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable (Cherlin, Shelah) and supersimple (Pillay, Scanlon, Wagner) ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed that a small field is algebraically closed , and asked whether a small field should be commutative. We shall answer this question positively in non-zero characteristic