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DP AND OTHER MINIMALITIES

Part of: Model theory

Published online by Cambridge University Press:  11 April 2025

PIERRE SIMON Open the ORCID record for PIERRE SIMON [Opens in a new window]  and
ERIK WALSBERG
PIERRE SIMON
Affiliation:
DEPARTMENT OF MATHEMATICS UC BERKELEY BERKELEY, CA 94720 USA E-mail: pierre.sim85@gmail.com
ERIK WALSBERG*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA IRVINE, IRVINE, CA 92697 USA
*
E-mail: erik.walsberg@gmail.com
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Abstract

A first-order expansion of $(\mathbb {R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, p-adic fields, ordered abelian groups with only finitely many convex subgroups (in particular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb {Z},+,S)$, where S is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb {Q},+,<)$ and o-minimal expansions ${\mathscr R}$ of $(\mathbb {R},+,<)$ such that $({\mathscr R},\mathbb {Q})$ is a “dense pair.”.

Keywords

dp-minimalmodel theorylogicordered structures

MSC classification

Primary: 03C45: Classification theory, stability and related concepts 03C60: Model-theoretic algebra 03C64: Model theory of ordered structures; o-minimality

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The Journal of Symbolic Logic , First View , pp. 1 - 30
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© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

REFERENCES

Alouf, E. and d’Elbée, C., A new dp-minimal expansion of the integers . Journal of Symbolic Logic, vol. 84 (2019), no. 2, pp. 632–663.10.1017/jsl.2019.15CrossRefGoogle Scholar
Aschenbrenner, M., Dolich, A., Haskell, D., Macpherson, D., and Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, I . Transactions of the American Mathematical Society, vol. 368 (2016), no. 8, pp. 5889–5949.CrossRefGoogle Scholar
Bar-Yehuda, E., Hasson, A., and Peterzil, Y., A theory of pairs for non-valuational structures . Journal of Symbolic Logic, vol. 84 (2019), no. 2, pp. 664–683.CrossRefGoogle Scholar
Bélair, L., Substructures and uniform elimination for $p$ -adic fields . Annals of Pure and Applied Logic, vol. 39 (1988), no. 1, pp. 1–17.CrossRefGoogle Scholar
Belegradek, O., Poly-regular ordered abelian groups , Logic and Algebra, volume 302 of Contemporary Mathematics (Y. Zhang, editor), American Mathematical Society, Providence, 2002, pp. 101–111.CrossRefGoogle Scholar
Belegradek, O. and Zilber, B., The model theory of the field of reals with a subgroup of the unit circle . Journal of the London Mathematical Society (2), vol. 78 (2008), no. 3, pp. 563–579.CrossRefGoogle Scholar
Berenstein, A. and Vassiliev, E., Geometric structures with a dense independent subset . Selecta Mathematica (N.S.), vol. 22 (2016), no. 1, pp. 191–225.CrossRefGoogle Scholar
Block Gorman, A., Hieronymi, P., and Kaplan, E., Pairs of theories satisfying a Mordell-Lang condition . Fundamenta Mathematicae, vol. 251 (2020), no. 2, pp. 131–160.10.4064/fm857-1-2020CrossRefGoogle Scholar
Chernikov, A. and Simon, P., Externally definable sets and dependent pairs . Israel Journal of Mathematics, vol. 194 (2013), no. 1, pp. 409–425.CrossRefGoogle Scholar
Cluckers, R., Presburger sets and $p$ -minimal fields . Journal of Symbolic Logic, vol. 68 (2003), no. 1, pp. 153–162.10.2178/jsl/1045861509CrossRefGoogle Scholar
Conant, G. and Pillay, A., Stable groups and expansions of $\left(\mathbb{Z},+,0\right)$ . Fundamenta Mathematicae, vol. 242 (2018), no. 3, pp. 267–279.CrossRefGoogle Scholar
Delon, F., Définissabilité avec paramètres extérieurs dans ${Q}_p$ et R. Proceedings of the American Mathematical Society, vol. 106 (1989), no. 1, pp. 193–198.Google Scholar
Dolich, A., Goodrick, J., and Lippel, D., Dp-minimality: basic facts and examples . Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.10.1215/00294527-1435456CrossRefGoogle Scholar
Edmundo, M. J., Structure theorems for o-minimal expansions of groups . Annals of Pure and Applied Logic, vol. 102 (2000), nos. 1–2, pp. 159–181.10.1016/S0168-0072(99)00043-3CrossRefGoogle Scholar
Engler, A. J. and Prestel, A., Valued Fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
Goodrick, J., A monotonicity theorem for DP-minimal densely ordered groups . Journal of Symbolic Logic, vol. 75 (2010), no. 1, pp. 221–238.10.2178/jsl/1264433917CrossRefGoogle Scholar
Günaydin, A. and Hieronymi, P., Dependent pairs . Journal of Symbolic Logic, vol. 76 (2011), no. 2, pp. 377–390.10.2178/jsl/1305810753CrossRefGoogle Scholar
Haskell, D. and Macpherson, D., Cell decompositions of C-minimal structures . Annals of Pure and Applied Logic, vol. 66 (1994), no. 2, pp. 113–162.10.1016/0168-0072(94)90064-7CrossRefGoogle Scholar
Haskell, D. and Macpherson, D., A version of o-minimality for the $p$ -adics . Journal of Symbolic Logic, vol. 62 (1997), no. 4, pp. 1075–1092.CrossRefGoogle Scholar
Hieronymi, P., The real field with an irrational power function and a dense multiplicative subgroup . Journal of the London Mathematical Society (2), vol. 83 (2011), no. 1, pp. 153–167.10.1112/jlms/jdq058CrossRefGoogle Scholar
Jahnke, F., Simon, P., and Walsberg, E., Dp-minimal valued fields . Journal of Symbolic Logic, vol. 82 (2017), no. 1, pp. 151–165.CrossRefGoogle Scholar
Johnson, W., The classification of dp-minimal and dp-small fields . J. Eur. Math. Soc., vol. 25 (2023), pp. 467–513.Google Scholar
Johnson, W., The canonical topology on dp-minimal fields . Journal of Mathematical Logic, vol. 18 (2018), no. 2, p. 1850007. 23.CrossRefGoogle Scholar
Johnson, W., Dp-finite fields II: The canonical topology and its relation to henselianity, preprint, 2019, arXiv:1910.05932.Google Scholar
Johnson, W., Dp-finite fields V: Topological fields of finite weight, preprint, 2020, arXiv:2004.14732.Google Scholar
Keren, G., Definable compactness in weakly o-minimal structures, Master’s thesis, Ben Gurion University of the Negev, 2014.Google Scholar
Laskowski, M. C. and Steinhorn, C., On o-minimal expansions of Archimedean ordered groups . Journal of Symbolic Logic, vol. 60 (1995) no. 3, pp. 817–831.CrossRefGoogle Scholar
Le Gal, O., A generic condition implying o-minimality for restricted ${C}^{\infty }$ -functions . Annales de la Faculté des Sciences de Toulouse: Mathématiques (6), vol. 19 (2010), nos. 3–4, pp. 479–492.Google Scholar
Macpherson, D., Marker, D., and Steinhorn, C., Weakly o-minimal structures and real closed fields . Transactions of the American Mathematical Society, vol. 352 (2000), no. 12, pp. 5435–5483.CrossRefGoogle Scholar
Marker, D. and Steinhorn, C. I., Definable types in o-minimal theories . Journal of Symbolic Logic, vol. 59 (1994), no. 1, pp. 185–198.10.2307/2275260CrossRefGoogle Scholar
Michaux, C. and Villemaire, R., Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham’s and Semenov’s theorems . Annals of Pure and Applied Logic, vol. 77 (1996), no. 3, pp. 251–277.CrossRefGoogle Scholar
Miller, C., Exponentiation is hard to avoid . Proceedings of the American Mathematical Society, vol. 122 (1994), no. 1, 257–259.CrossRefGoogle Scholar
Miller, C. and Starchenko, S., Status of the o-minimal two group question, 2003. Available at https://people.math.osu.edu/miller.1987/tg.pdf.Google Scholar
Peterzil, Y., Pillay’s conjecture and its solution—A survey , Logic Colloquium 2007 (F. Delon, U. Kohlenbach, P. Maddy, and F. Stephan, editors), volume 35 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2010, pp. 177–203.CrossRefGoogle Scholar
Peterzil, Y. and Starchenko, S., A trichotomy theorem for o-minimal structures . Proceedings of the London Mathematical Society (3), vol. 77 (1998), no. 3, pp. 481–523.10.1112/S0024611598000549CrossRefGoogle Scholar
Point, F. and Wagner, F. O., Essentially periodic ordered groups . Annals of Pure and Applied Logic, vol. 105 (2000), nos. 1–3, pp. 261–291.10.1016/S0168-0072(99)00053-6CrossRefGoogle Scholar
Prestel, A. and Roquette, P., Formally $\boldsymbol{p}$ -Adic Fields, volume 1050 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1984.Google Scholar
Robinson, A. and Zakon, E., Elementary properties of ordered abelian groups . Transactions of the American Mathematical Society, vol. 96 (1960), pp. 222–236.10.1090/S0002-9947-1960-0114855-0CrossRefGoogle Scholar
Shelah, S., Strongly dependent theories . Israel Journal of Mathematics, vol. 204 (2014), no. 1, pp. 1–83.CrossRefGoogle Scholar
Simon, P., On dp-minimal ordered structures . Journal of Symbolic Logic, vol. 76 (2011), no. 2, pp. 448–460.10.2178/jsl/1305810758CrossRefGoogle Scholar
Simon, P., A Guide to NIP Theories, volume 44 of Lecture Notes in Logic, Cambridge University Press, 2015.10.1017/CBO9781107415133CrossRefGoogle Scholar
Simon, P. and Walsberg, E., Tame topology over dp-minimal structures . Notre Dame Journal of Formal Logic, vol. 60 (2019), no. 1, pp. 61–76.10.1215/00294527-2018-0019CrossRefGoogle Scholar
Świerczkowski, S., On cyclically ordered groups . Fundamenta Mathematicae, vol. 47 (1959), pp. 161–166.CrossRefGoogle Scholar
Tran, M. C. and Walsberg, E., A family of dp-minimal expansions of $\left(\mathbb{Z};+\right)$ . Notre Dame J. Form. Log., vol. 64 (2023), no. 2, 225–238.Google Scholar
Tychonievich, M. A., Defining additive subgroups of the reals from convex subsets . Proceedings of the American Mathematical Society, vol. 137 (2009), no. 10, pp. 3473–3476.CrossRefGoogle Scholar
Wagner, F. O., Bad fields in positive characteristic . Bulletin of the London Mathematical Society, vol. 35 (2003), no. 4, pp. 499–502.10.1112/S0024609303001929CrossRefGoogle Scholar
Walsberg, E., An $NIP$ structure which does not interpret an infinite group but whose shelah expansion interprets an infinite field, preprint, 2019, arXiv:1910.13504.Google Scholar
Wencel, R., Weakly o-minimal nonvaluational structures . Annals of Pure and Applied Logic, vol. 154 (2008), no. 3, pp. 139–162.10.1016/j.apal.2008.01.009CrossRefGoogle Scholar
Wencel, R., On the strong cell decomposition property for weakly o-minimal structures . Mathematical Logic Quarterly, vol. 59 (2013), no. 6, pp. 452–470.10.1002/malq.201200016CrossRefGoogle Scholar
Wilkie, A. J., Fusing o-minimal structures . Journal of Symbolic Logic, vol. 70 (2005), no. 1, pp. 271–281.CrossRefGoogle Scholar