Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Some model theory of Polish structures

Author(s): Krzysztof Krupinski
Journal: Trans. Amer. Math. Soc. 362 (2010), 3499-3533.
MSC (2010): Primary 03C45, 03E15; Secondary 54H11, 20E18, 54F15
Posted: February 15, 2010
MathSciNet review: 2601598
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We introduce a notion of Polish structure and, in doing so, provide a setting which allows the application of ideas and techniques from model theory, descriptive set theory, topology and the theory of profinite groups. We define a topological notion of independence in Polish structures and prove that it has some nice properties. Using this notion, we prove counterparts of some basic results from geometric stability theory in the context of small Polish structures. Then, we prove some structural theorems about compact groups regarded as Polish structures: each small, -stable compact -group is solvable-by-finite; each small compact -group of finite ${\mathcal NM}$ -rank is nilpotent-by-finite. Examples of small Polish structures and groups are also given.

References:

1.: A. V. Arhangel'skii (Ed.), General Topology III, Springer, Berlin, Heidelberg, 1995. MR 1416131 (97f:54001)
2.: H. Becker, A. S. Kechris, The Descriptive Set Theory of Polish Groups Actions, Cambridge University Press, Cambridge, 1996. MR 1425877 (98d:54068)
3.: R. Engelking, Topologia Ogólna, PWN, Warszawa, 1989.
4.: C. Ealy, K. Krupiński, A. Pillay, Superrosy groups having finitely satisfiable generics, Ann. Pure Appl. Logic 151 (2008), 1-21. MR 2381504 (2009a:03041)
5.: E. Hewitt, K. Ross, Abstract Harmonic Analysis II, Springer, Berlin, 1970. MR 0262773 (41:7378)
6.: G. Hjorth, A universal Polish -space, Topology Appl. 91 (1999), 141-150. MR 1664504 (2000a:54016)
7.: A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995. MR 1321597 (96e:03057)
8.: O. Kegel, B. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam, 1973. MR 0470081 (57:9848)
9.: J. Kennedy, J. T. Rogers, Jr., Orbits of the pseudocircle, Trans. Amer. Math. Soc. 296 (1986), 327-340. MR 837815 (87g:54076)
10.: K. Krupiński, Products of finite abelian groups as profinite groups, J. Alg. 288 (2005), 556-582. MR 2146145 (2006g:20042)
11.: -, Abelian profinite groups, Fund. Math. 185 (2005), 41-59. MR 2161751 (2006k:20052)
12.: -, Profinite structures interpretable in fields, Ann. Pure Appl. Logic 142 (2006), 19-54. MR 2250536 (2007k:03090)
13.: -, Generalizations of small profinite structures, Submitted.
14.: -, Fields interpretable in rosy theories, Israel J. Math., to appear.
15.: -, F. O. Wagner, Small profinite groups and rings, J. Alg. 306 (2006), 494-506. MR 2271348 (2008c:20052)
16.: G. L. Lehner, Extending homeomorphisms on the pseudo-arc, Trans. Amer. Math. Soc. 98 (1961), 369-394. MR 0120608 (22:11358)
17.: M. Megrelishvili, Free topological -groups, New Zeland J. Math. 25 (1996), 59-72. MR 1398366 (97f:54047)
18.: J. van Mill, Infinite-Dimensional Topology. Prerequisites and Introduction, Elsevier Science Publishers, Amsterdam, 1989. MR 977744 (90a:57025)
19.: S. B. Nadler, Jr., Continuum Theory. An Introduction, Marcel Dekker, New York, 1992. MR 1192552 (93m:54002)
20.: L. Newelski, ${\mathcal M}$ -gap conjecture and -normal theories, Israel J. Math. 106 (1998), 285-311. MR 1656901 (2000b:03122)
21.: -, Small profinite groups, J. Symb. Logic 66 (2001), 859-872. MR 1833483 (2002h:20040)
22.: -, Small profinite structures, Trans. Amer. Math. Soc. 354 (2002), 925-943. MR 1867365 (2002j:03038)
23.: A. Pillay, Geometric Stability Theory, Oxford University Press, New York, 1996. MR 1429864 (98a:03049)
24.: B. Poizat, Stable Groups, American Mathematical Society, Providence, 2001. MR 1827833 (2002a:03067)
25.: L. Ribes, P. Zalesskii, Profinite Groups, Springer, Berlin, 2000. MR 1775104 (2001k:20060)
26.: F. O. Wagner, Simple Theories, Kluwer Academic Publishers, Dordrecht, 2000. MR 1747713 (2001b:03035)
27.: -, Small profinite -stable groups, Fund. Math. 176 (2003), 181-191. MR 1971308 (2004d:20029)
28.: J. S. Wilson, On the structure of compact torsion groups, Monatsh. Math. 96 (1983), 57-66. MR 721596 (85a:22007)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|