Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T03:57:02.126Z Has data issue: false hasContentIssue false
  • English
  • Français

The last word on elimination of quantifiers in modules

Published online by Cambridge University Press:  12 March 2014

Hans B. Gute  and
K. K. Reuter
Hans B. Gute
Affiliation:
Équipe de Logique Mathématique, Université Paris-VII, 75251 Paris, France
K. K. Reuter
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität, 7800 Freiburg, West Germany
Get access Rights & Permissions [Opens in a new window]

Extract

In what follows, a coset is a subset of a group G of the form aH, where H is a subgroup of G; H can be recovered from the coset C: it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka, since aH = aHa−1a. A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C1 = a1H1,…,Cn = anHn, then it is the union of those Ci whose corresponding Hi has finite index in H.

In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power Gn is a(finite!) Boolean combination of cosets modulo definable subgroups. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990 , pp. 670 - 673
Copyright
Copyright © Association for Symbolic Logic 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Baur, W. [1976], Elimination of quantifiers for modules, Israel Journal of Mathematics, vol. 25, pp. 64–70.CrossRefGoogle Scholar
Hrushovski, E. and Pillay, A. [1987], Weakly normal groups, Logic Colloquium '85, North-Holland, Amsterdam, 1987, pp. 233–244.CrossRefGoogle Scholar
Neumann, B. H. [1952], A note on algebraically closed groups, Journal of the London Mathematical Society, vol. 27, pp. 247–249.CrossRefGoogle Scholar
Poizat, B. [1985], Cours de théorie des modèles, Nur al-Mantiq wal-Ma'rifah, Villeurbanne.Google Scholar
Poizat, B. [1987], À propos de groupes stables, Logic Colloquium '85, North-Holland, Amsterdam, 1987, pp. 245–265.CrossRefGoogle Scholar
Szmielew, W. [1955], Elementary properties of abelian groups, Fundamenta Mathematicae, vol. 41, pp. 203–271.CrossRefGoogle Scholar
Ziegler, M. [1984], Model theory of modules, Annals of Pure and Applied Logic, vol. 26, pp. 149–213.CrossRefGoogle Scholar