Spectral Spaces in o-minimal and other NIP theories
| dc.contributor.author | Baro González, Elías | |
| dc.contributor.author | Fernando Galván, José Francisco | |
| dc.contributor.author | Palacín Cruz, Daniel | |
| dc.date.accessioned | 2023-06-22T12:29:12Z | |
| dc.date.available | 2023-06-22T12:29:12Z | |
| dc.date.issued | 2022-08-02 | |
| dc.description.abstract | We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the honest topology. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon’s FM retraction. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | FALSE | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovación | |
| dc.description.sponsorship | Comunidad de Madrid | |
| dc.description.sponsorship | Universidad Complutense de Madrid | |
| dc.description.status | unpub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/75532 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/72650 | |
| dc.language.iso | eng | |
| dc.relation.projectID | STRANO (PID2021-122752NB-I00) ; STRANO (MTM2017-82105-P) | |
| dc.relation.projectID | 2020-T1/TIC-20313 | |
| dc.relation.projectID | UCM (910444) | |
| dc.rights | Atribución 3.0 España | |
| dc.rights.accessRights | open access | |
| dc.rights.uri | https://creativecommons.org/licenses/by/3.0/es/ | |
| dc.subject.cdu | 510.67 | |
| dc.subject.cdu | 515.122 | |
| dc.subject.ucm | Lógica simbólica y matemática (Matemáticas) | |
| dc.subject.ucm | Topología | |
| dc.subject.unesco | 1102.14 Lógica Simbólica | |
| dc.subject.unesco | 1210 Topología | |
| dc.title | Spectral Spaces in o-minimal and other NIP theories | |
| dc.type | journal article | |
| dcterms.references | [1] E. Baro, Jose F. Fernando and J.M. Gamboa. Spectral maps associated to semialgebraic branched coverings. Rev. Mat. Complut. 35 (2022), no. 1, 227–264. [2] A. Berarducci and A. Fornasiero. o-minimal cohomology: finiteness and invariance results. J. Math. Log. 9 (2009), no. 2, 167–182. [3] J. Bochnak, M. Coste and M.-F. Roy. Real Algebraic Geometry. Ergeb. Math. Grenzgeb., vol. 36, Springer-Verlag, Berlin, 1998. [4] M. Carral and M. Coste. Normal spectral spaces and their dimensions. J. Pure and Appl. Algebra 30 (1983), 227–235. [5] M. Coste and M-F. Roy. La topologie du spectre réel. Ordered fields and real algebraic geometry (San Francisco, Calif., 1981), pp. 27–59, Contemp. Math., 8, Amer. Math. Soc., Providence, R.I., 1982. [6] A. Chernikov and P. Simon. Externally definable sets and dependent pairs. Israel J. Math. 194 (2013), 409–425. [7] A. Chernikov, A. Pillay and P. Simon. External definability and groups in NIP theories. J. Lond. Math. Soc. (2) 90 (2014), no. 1, 213–240. [8] M. Dickmann, N. Schwartz and M. Tressl. Spectral spaces. New Mathematical Monographs, 35. Cambridge University Press, Cambridge, 2019. [9] A. Dolich. Forking and independence in o-minimal theories. J. Symb. Logic 69 (2004), 215– 240. [10] M. Edmundo, G. Jones and N. Peatfield. Sheaf cohomology in o-minimal structures. J. Math. Log. 6 (2006), no. 2, 163–179. [11] M. Edmundo and L. Prelli. The six Grothendieck operations on o-minimal sheaves. Math. Z. 294 (2020), no. 1-2, 109–160. [12] L. van den Dries. Tame topoloy and o-minimal structures. London Math. Soc., Lecture Note Series 248, Cambridge University Press, 1998. [13] J. F. Fernando and J.M. Gamboa, On the semialgebraic Stone-Čech compactification of a semialgebraic set. Trans. Amer. Math. Soc. 364 (2012), no. 7, 3479–3511. [14] J. F. Fernando and J. M. Gamboa. On Łojasiewicz’s inequality and the Nullstellensatz for rings of semialgebraic functions. J. Algebra 399 (2014), 475–488. [15] A. Fornasiero. O-minimal spectrum. arXiv: https://arxiv.org/pdf/math/0612366v1.pdf [16] J. Gismatullin, D. Penazzi and A. Pillay. On compactifications and the topological dynamics of definable groups. Ann. Pure Appl. Logic 165, 552–562 (2014). [17] E. Hrushovski and A. Pillay. On NIP and invariant measures. J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1005–1061. [18] Y. Peterzil and A. Pillay. Generic sets in definably compact groups. Fund. Math. 193 (2007), 153–170. [19] K. Peterzil and S. Starchenko. Topological groups, μ-types and their stabilizers. J. European Math. Soc. (2017), 2965–2995. [20] A. Pillay. First Order Topological Structures and Theories. J. Symb. Logic 52 (1987), 763–778. [21] A. Pillay. On groups and rings definable in o-minimal structures. J. Pure Appl. Algebra 53 (1988), 239–255. [22] A. Pillay, Sheaves of Continuous Definable Functions. J. Symb. Logic 53 (1988), 1165–1169. [23] P. Simon. Invariant types in NIP theories. J. Math. Log. 15 (2015), 26 pp. [24] P. Simon and E. Walsberg. Tame topology over dp-minimal structures. Notre Dame J. Form. Log. 60 (2019), no. 1, 61–76. [25] S. Starchenko. A note on Dolich’s paper. Modnet Preprint server 131 (2008), http://www. logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/131/131.pdf [26] P. Simon and S. Starchenko. On forking and definability of types in some dp-minimal theories. J. Symb. Log. 79 (2014), no. 4, 1020–1024. [27] K. Tent and M. Ziegler. A Course in Model Theory. Lecture Notes in Logic 40, Cambridge University Press, 2012. [28] M. Tressl, The real spectrum of continuous definable functions in o-minimal structures. Séminaire de Structures Algébriques Ordonnées 1997-1998 (1999), 1–15. | |
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