Colloque en l'honneur de Michèle Giraudet

Paris - Chevaleret, 27 - 28 novembre 2006


A propos de groupes et de structures ordonnées
Around Groups and Ordered Structures

Résumés



Françoise Delon
Équipe de Logique Mathématique
CNRS - Université Paris Diderot

Groupes valués C-minimaux abéliens.

La minimalité forte a inspiré d'autres notions de minimalité, adaptées au cas instable. Ainsi de l'o-minimalité, en présence d'un ordre linéaire ou de la C-minimalité, en présence d'une ultramétrique. Nous avons entamé avec Patrick Simonetta une classification des groupes valués C-minimaux abéliens, dont je présenterai les premiers résultats.



Manfred Droste
Universität Leipzig
Institut für Informatik

Normal subgroups of unital automorphism groups (Joint work with Charles Holland)

We investigate the group of all those order automorphisms of the reals which are bounded by some power of translation by 1. This is a unital lattice-ordered group which has independent interest due to its recent connections to mv-algebras. It is well known that the group of all order automorphisms of the reals has just three proper non-trivial normal subgroups. In contrast, we show that our group has 2c many normal subgroups, where c denotes the cardinality of the continuum. We give a complete description of this normal subgroup lattice.



Andrew Glass
University of Cambridge

Automorphism Groups



Rüdiger Göbel
Universität Duisburg - Essen

On Shelah's absolute trees and applications

In 1982 (Israel Journal) Saharon Shelah showed that the first Erdos cardinal λ is the exact upper bound for which we can find families of size < λ of absolute rigid trees. Rigid means that there is no color preserving homomorphism between any two distinct members, and absolute means that this property is preserved when changing the universe (as explained in set theory).

Shelah's trees can be encoded into K-vector spaces V with 5 distinguished subspaces to show that there are absolute rigid families of vector spaces iff their dimension is < λ. Here rigid means that End V=K and there are no K-homomorphisms between any two distinct members. This is joint work with Shelah (to appear Proc. AMS), which closes a gap in a paper by Eklof and Shelah (and in the new edition of the book by Eklof and Mekler).

The result can be extended to R-modules for a large class of commutative rings (joint work with Fuchs). In particular it gives a new construction of abelian groups A with End A = Z for |A|< λ.

There is a parallel result on endomorphism monoids of graphs (joint work with Droste and Pokutta).



Charles Holland
Bowling Green State University

Non-commutative logic and lattice-ordered groups

Because of our quantum world, there is some reason to suppose that the statements "x and y" and "y and x" might not have the same truth value. The associated logic must then be multi-valued, and the corresponding algebra of propositions is intimately connected with lattice-ordered groups.



Anatole Khélif
Équipe de Logique Mathématique
CNRS - Université Paris Diderot

Groupes QFA ordonnés

La notion de groupe QFA (Quasi-Finiment Axiomatisable) a été introduite par Nies. Il est bien connu que (Z,+) n'est pas QFA. Mais (Z,+) devient QFA si on y ajoute l'ordre.
Nous nous proposons de montrer que tout groupe nilpotent de type fini sans torsion peut être muni d'un ordre qui le rende QFA et nous donnerons un exemple de groupe abélien ordonné de type fini non QFA.



François Lucas
Université d'Angers

Théorie des modèles des groupes cycliquement ordonnés (Travail commun avec Michèle Giraudet)

On rappelle les résultats classiques, théorème de représentation de Rieger et théorème de plongement et on en donne des analogues modèle-théorétiques, On montre que les g.c.ordonnables forment une classe élémentaire, et on s'intéresse aux théories de g.c.o. abéliens.



Françoise Point
F.N.R.S. - Université de Mons-Hainaut

On the theories of finitely presented lattice-ordered abelian groups.

First, I will recall a result obtained in collaboration with A. Glass and A. Macintyre on the classification of finitely generated free abelian lattice ordered groups according to their number of generators. Then, I will describe some partial results on the theories of finitely presented lattice-ordered abelian groups (joint work with A. Glass).



J. K. Truss
University of Leeds

Countable homogeneous coloured partial orders (Joint work with Susana Torrezao de Sousa)

We give a classification of all the countable homogeneous coloured partial orders. This generalizes the similar result in the monochromatic case given by Schmerl.