Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-25T05:44:42.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizations of small profinite structures

Published online by Cambridge University Press:  12 March 2014

Krzysztof Krupiński
Krzysztof Krupiński*
Affiliation:
Instytut Matematyczny Uniwersytetu Wrocławskiego, Pl. Grunwaldzki 2/4. 50-384 Wrocław, Poland Mathematics Department, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801., USA. E-mail: kkrup@math.uni.wroc.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms and satisfying the existence of m-independent extensions (we call them compact e-structures). We analyze the relationships between smallness and different versions of the assumption of the existence of m-independent extensions and we obtain some topological consequences of these assumptions. Using them, we adopt Newelski's proofs of various results about small profinite structures to compact e-structures. In particular, we notice that a variant of the group configuration theorem holds in this context.

A general construction of compact structures is described. Using it, a class of examples of compact e-structures which are not small is constructed.

It is also noticed that in an m-stable compact e-structure every orbit is equidominant with a product of m-regular orbits.

Keywords

Profinite structurecompact structureindependence relation
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 4 , December 2010 , pp. 1147 - 1175
Copyright
Copyright © Association for Symbolic Logic 2010

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

[1]Casanovas, E., Lascar, D., Pillay, A., and Ziegler, M., Galois groups of first order theories, Journal of Mathematical Logic, vol. 1 (2001), pp. 305–319.CrossRefGoogle Scholar
[2]Wagner, F.Krupiński, K., Small profinite groups and rings, Journal of Algebra, vol. 306 (2006), pp. 494–506.Google Scholar
[3]Kim, B., A note on Lascar strong types in simple theories, this Journal, vol. 63 (2000), pp. 926–936.Google Scholar
[4]Krupiński, K., Abelian profinite groups, Fundamenta Mathematicae, vol. 185 (2005), pp. 41–59.CrossRefGoogle Scholar
[5]Krupiński, K., Products of finite abelian groups as profinite groups, Journal of Algebra, vol. 288 (2005), pp. 556–582.CrossRefGoogle Scholar
[6]Krupiński, K., Profinite structures interpretable in fields, Annals of Pure and Applied Logic, vol. 142 (2006), pp. 19–54.CrossRefGoogle Scholar
[7]Krupiński, K. and Newelski, L., On bounded type-definable equivalence relations, Notre Dame Journal of Formal Logic, vol. 43 (2002), pp. 231–242.CrossRefGoogle Scholar
[8]Lascar, D. and Pillay, A., Hyperimaginaries and automorphism groups, this Journal, vol. 66 (2001), pp. 127–143.Google Scholar
[9]Newelski, L., ℳ-gap conjecture and m-normal theories, Israel Journal of Mathematics, vol. 106 (1998), pp. 285–311.CrossRefGoogle Scholar
[10]Newelski, L., Flat Morley sequences, this Journal, vol. 64 (1999), pp. 1261–1279.Google Scholar
[11]Newelski, L., Geometry of *-finite types, this Journal, vol. 64 (1999), pp. 1375–1395.Google Scholar
[12]Newelski, L., m-normal theories, Fundamenta Mathematicae, vol. 170 (2001), pp. 141–163.CrossRefGoogle Scholar
[13]Newelski, L., Small profinite groups, this Journal, vol. 66 (2001), pp. 859–872.Google Scholar
[14]Newelski, L., Small profinite structures, Transactions of the American Mathematical Society, vol. 354 (2002), pp. 925–943.CrossRefGoogle Scholar
[15]Wagner, F. O., Simple theories, Kluwer Academic Publishers. 2000.CrossRefGoogle Scholar
[16]Wagner, F. O., Small profinite m-stable groups, Fundamenta Mathematicae, vol. 176 (2003), pp. 181–191.CrossRefGoogle Scholar