Canonical forests in directed families
Authors:
Joseph Flenner and Vincent Guingona
Journal:
Proc. Amer. Math. Soc. 142 (2014), 1849-1860
MSC (2010):
Primary 06A07, 03C45.
DOI:
https://doi.org/10.1090/S0002-9939-2014-11935-6
Published electronically:
February 26, 2014
MathSciNet review:
3182006
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Two uniqueness results on representations of sets constructible in a directed family of sets are given. In the unpackable case, swiss cheese decompositions are unique. In the packable case, they are not unique but admit a quasi-ordering under which the minimal decomposition is unique. Both cases lead to a one-dimensional elimination of imaginaries in VC-minimal and quasi-VC-minimal theories.
- Hans Adler, A geometric introduction to forking and thorn-forking, J. Math. Log. 9 (2009), no. 1, 1–20. MR 2665779, DOI https://doi.org/10.1142/S0219061309000811
- Sarah Cotter and Sergei Starchenko, Forking in VC-minimal theories, J. Symbolic Logic 77 (2012), no. 4, 1257–1271. MR 3051624, DOI https://doi.org/10.2178/jsl.7704110
- Joseph Flenner and Vincent Guingona, Convexly orderable groups and valued fields, to appear in Journal of Symbolic Logic.
- Vincent Guingona, Computing VC-density in VC-minimal theories, in preparation.
- Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson, Definable sets in algebraically closed valued fields: elimination of imaginaries, J. Reine Angew. Math. 597 (2006), 175–236. MR 2264318, DOI https://doi.org/10.1515/CRELLE.2006.066
- Jan E. Holly, Canonical forms for definable subsets of algebraically closed and real closed valued fields, J. Symbolic Logic 60 (1995), no. 3, 843–860. MR 1348997, DOI https://doi.org/10.2307/2275760
- Jan E. Holly, Prototypes for definable subsets of algebraically closed valued fields, J. Symbolic Logic 62 (1997), no. 4, 1093–1141. MR 1618005, DOI https://doi.org/10.2307/2275629
- Bruno Poizat, A course in model theory, Universitext, Springer-Verlag, New York, 2000. An introduction to contemporary mathematical logic; Translated from the French by Moses Klein and revised by the author. MR 1757487
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 06A07, 03C45.
Retrieve articles in all journals with MSC (2010): 06A07, 03C45.
Additional Information
Joseph Flenner
Affiliation:
Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication:
Department of Mathematics, University of Saint Francis, 2701 Spring Street, Fort Wayne, Indiana 46808
Email:
jflenner@sf.edu
Vincent Guingona
Affiliation:
Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556
MR Author ID:
942387
Email:
guingona.1@nd.edu
Keywords:
Directed family of sets,
swiss cheese decomposition,
VC-minimality,
elimination of imaginaries
Received by editor(s):
November 10, 2011
Received by editor(s) in revised form:
June 29, 2012
Published electronically:
February 26, 2014
Additional Notes:
Both authors were supported by NSF grant DMS-0838506.
Communicated by:
Julia Knight
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
