zbMATH — the first resource for mathematics

Definable quotients of locally definable groups. (English) Zbl 1273.03130
Summary: We study locally definable abelian groups \({\mathcal{U}}\) in various settings and examine conditions under which the quotient of \({\mathcal{U}}\) by a discrete subgroup might be definable. This turns out to be related to the existence of the type-definable subgroup \({\mathcal{U}^{00}}\) and to the divisibility of \({\mathcal{U}}\).

03C64 Model theory of ordered structures; o-minimality
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
20A15 Applications of logic to group theory
Full Text: DOI arXiv
[1] Baro E., Edmundo M.J.: Corrigendum to: ”Locally definable groups in o-minimal structures” by Edmundo. J. Algebra 320(7), 3079–3080 (2008)
[2] Baro E., Otero M.: Locally definable homotopy. Ann. Pure Appl. Log. 161(4), 488–503 (2010) · Zbl 1225.03043
[3] Berarducci A., Mamino M.: On the homotopy type of definable groups in an o-minimal structure. J. Lond. Math. Soc. (2) 83(3), 563–586 (2011) · Zbl 1225.03044
[4] Berarducci A., Otero M., Peterzil Y., Pillay A.: A descending chain condition for groups definable in o-minimal structures. Ann. Pure Appl. Log. 134(2–3), 303–313 (2005) · Zbl 1068.03033
[5] Dolich A.: Forking and independence in o-minimal theories. J. Symbol. Log. 69, 215–240 (2004) · Zbl 1074.03016
[6] Edmundo M.J.: Locally definable groups in o-minimal structures. J. Algebra 301(1), 194–223 (2006) · Zbl 1104.03032
[7] Edmundo M.J., Eleftheriou P.E.: The universal covering homomorphism in o-minimal expansions of groups. Math. Log. Q. 53, 571–582 (2007) · Zbl 1130.03027
[8] Eleftheriou, P.E., Peterzil, Y.: Definable groups as homomorphic images of semilinear and field-definable groups. doi: 10.1007/S00029-012-0092-4 · Zbl 1273.03129
[9] Eleftheriou, P.E., Peterzil, Y.:Lattices in locally definable subgroups of $${{\(\backslash\)langle}R\^{n}, +{\(\backslash\)rangle}}$$ . preprint · Zbl 1345.03072
[10] Eleftheriou P.E., Starchenko S.: Groups definable in ordered vector spaces over ordered division rings. J. Symb. Log. 72(4), 1108–1140 (2007) · Zbl 1130.03028
[11] Hofmann, K.H., Morris, S.A.: The Structure of Compact Groups. A Primer for the Student–A Handbook for the Expert. de Gruyter Studies in Mathematics, vol. 25. Walter de Gruyter & Co., Berlin (1998) · Zbl 0919.22001
[12] Hrushovski E., Peterzil Y., Pillay A.: Groups, measures, and the NIP. J. Am. Math. Soc. 21(2), 563–596 (2008) · Zbl 1134.03024
[13] Komjáth P., Totik V.: Problems and Theorems in Classical Set Theory, Problem Books in Mathematics. Springer, New York (2006) · Zbl 1103.03041
[14] Peterzil Y.: Returning to semi-bounded sets. J. Symb. Logic 74(2), 597–617 (2009) · Zbl 1171.03022
[15] Peterzil Y., Starchenko S.: A trichotomy theorem for ominimal structures. Proc. Lond. Math. Soc. 77(3), 481–523 (1998) · Zbl 0904.03021
[16] Peterzil Y., Pillay A.: Generic sets in definably compact groups. Fund. Math. 193(2), 153–170 (2007) · Zbl 1117.03042
[17] Pillay A.: Type-definability, compact Lie groups, and o-minimality. J. Math. Log. 4(2), 147–162 (2004) · Zbl 1069.03029
[18] Shelah S.: Minimal bounded index subgroups and dependent theories. Proc. Am. Math. Soc. 136, 1087–1091 (2008) · Zbl 1144.03026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.