Canonical forests in directed families

Flenner, Joseph; Guingona, Vincent

doi:10.1090/S0002-9939-2014-11935-6

Canonical forests in directed families
HTML articles powered by AMS MathViewer

by Joseph Flenner and Vincent Guingona PDF

Proc. Amer. Math. Soc. 142 (2014), 1849-1860 Request permission

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.

References

Hans Adler, A geometric introduction to forking and thorn-forking, J. Math. Log. 9 (2009), no. 1, 1–20. MR 2665779, DOI 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 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 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 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 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, DOI 10.1007/978-1-4419-8622-1