In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
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Adler, H.: Theories controlled by formulas of Vapnik-Chervonenkis codimension 1. Preprint (2008)
Aschenbrenner, A., Dolich, A., Haskell, D., MacPherson, H.D., Starchenko, S.: Vapnik-Chervonenkis density in some theories without the independence property, II. Notre Dame J. Form. Log. 54(3–4), 311–363 (2013)
Cotter S., Starchenko S.: Forking in VC-minimal theories. J. Symb. Log. 77(4), 1257–1271 (2012)
Dolich A., Goodrick J., Lippel D.: Dp-minimality: Basic facts and examples. Notre Dame J. Form. Log. 52(3), 267–288 (2011)
Flenner, J., Guingona, V.: Canonical forests in directed families. Proc. Am. Math. Soc. (2013) (to appear)
Flenner, J., Guingona, V.: Convexly orderable groups and valued fields. J. Symbolic Logic (2012) (to appear)
Goodrick J.: A monotonicity theorem for dp-minimal densely ordered groups. J. Symb. Log. 75(1), 221–238 (2010)
Guingona V., Laskowski M.C.: On VC-minimal theories and variants. Arch. Math. Log. 52(7), 743–758 (2013)
Krupinski K., Pillay A.: On stable fields and weight. J. Inst. Math. Jussieu 10(2), 349–358 (2011)
Macintyre A., McKenna K., Van den Dries L.: Elimination of quantifiers in algebraic structures. Adv. Math. 47, 74–87 (1983)
Marker D., MacPherson D., Steinhorn C.: Weakly o-minimal structures and real closed fields. Trans. Am. Math. Soc. 352(12), 5435–5483 (2000)
Onshuus A., Usvyatsov A.: On dp-minimality, strong dependence, and weight. J. Symb. Log. 76(3), 737–758 (2011)
Ribenboim, P.: Theorie des Valuations, vol. 40, pp. 2670. Les Presses de l’Universite de Montreal, Montreal (1964)
Simon P.: On dp-minimal ordered structures. J. Symb. Log. 76(2), 448–460 (2011)
Wood C.: Notes on the stability of separably closed fields. J. Symb. Log. 44, 412–416 (1979)
The author was supported by NSF Grant DMS-0838506.
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Guingona, V. On VC-minimal fields and dp-smallness. Arch. Math. Logic 53, 503–517 (2014). https://doi.org/10.1007/s00153-014-0376-9