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Non-standard lattices and o-minimal groups

Published online by Cambridge University Press:  05 September 2014

Pantelis E. Eleftheriou
Pantelis E. Eleftheriou*
Affiliation:
University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada E-mail: pelefthe@uwaterloo.ca
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Abstract

We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.

Keywords

03C6403C68O-minimalitynon-standard lattices-definable groups
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 19 , Issue 1 , March 2013 , pp. 56 - 76
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Baro, E. and Edmundo, M., Corrigendum to: “Locally definable groups in o-minimal structures” by Edmundo, Journal of Algebra, vol. 320 (2008), no. 7, pp. 3079–3080.Google Scholar
[2] Baro, E. and Otero, M., Locally definable homotopy, Annals of Pure and Applied Logic, vol. 161 (2010), no. 4, pp. 488–503.Google Scholar
[3] Berarducci, A., Edmundo, M., and Mamino, M., Discrete subgroups of locally definable groups, preprint, 2012.CrossRefGoogle Scholar
[4] Berarducci, A.. Otero, M., Peterzil, Y., and Pillay, A., A descending chain condition for groups definable in o-minimal structures, Annals of Pure and Applied Logic, vol. 134 (2005), pp. 303–313.Google Scholar
[5] Berenstein, A., Ealy, C., and Günaydin, A., Thorn independence in the field of real numbers with a small multiplicative group, Annals of Pure and Applied Logic, vol. 150 (2007), pp. 1–18.Google Scholar
[6] Dolich, A., Marker, D., and Steinhorn, C., Structures having o-minimal open core, Proceedings of the American Mathematical Society, vol. 139 (2011), pp. 319–330.Google Scholar
[7] van den Dries, L., Dense pairs of o-minimal structures, Fundamenta Mathematicae, vol. 157 (1998), no. 1, pp. 61–78.Google Scholar
[8] van den Dries, L., Tame topology and o-minimal structures, Cambridge University Press, Cambridge, 1998.Google Scholar
[9] van den Dries, L. and Günaydin, A., The fields of real and complex numbers with a small multiplicative group, Proceedings of the London Mathematical Society, vol. 93 (2006), pp. 43–81.Google Scholar
[10] Edmundo, M., Structure theorems for o-minimal expansions of groups, Annals of Pure and Applied Logic, vol. 102 (2000), pp. 159–181.Google Scholar
[11] Edmundo, M., Locally definable groups in o-minimal structures, Journal of Algebra, vol. 301 (2006), no. 1, pp. 194–223.Google Scholar
[12] Edmundo, M. and Eleftheriou, P., The universal covering homomorphism in o-minimal expansions of groups, Mathematical Logic Quarterly, vol. 53 (2007), pp. 571–582.Google Scholar
[13] Edmundo, M., Jones, G., and Peatfield, N., Sheaf cohomology in o-minimal structures, Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 163–179.Google Scholar
[14] Edmundo, M. and Otero, M., Definably compact abelian groups, Journal of Mathematical Logic, vol. 4 (2004), pp. 163–180.Google Scholar
[15] Edmundo, M. and Peatfield, N., O-minimal Cech cohomology, Quarterly Journal of Mathematics, vol. 59 (2008), no. 2, pp. 213–220.Google Scholar
[16] Edmundo, M. and Woerheide, A., Comparison theorems for o-minimal singular (co) homology, Transactions of the American Mathematical Society, vol. 360 (2008), pp. 4889–4912.Google Scholar
[17] Eleftheriou, P., A semi-linear group which is not affine, Annals of Pure and Applied Logic, vol. 156 (2008), pp. 287–289.CrossRefGoogle Scholar
[18] Eleftheriou, P., Compact domination for groups definable in linear o-minimal structures, Archive for Mathematical Logic, vol. 48 (2009), no. 7, pp. 607–623.Google Scholar
[19] Eleftheriou, P., Groups definable in linear o-minimal structures: the non-compact case, The Journal of Symbolic Logic, vol. 75 (2010), pp. 208–220.Google Scholar
[20] Eleftheriou, P., Local analysis for semi-bounded groups, Fundamenta Mathematicae, vol. 216 (2012), pp. 223–258.Google Scholar
[21] Eleftheriou, P. and Peterzil, Y., Definable groups as homomorphic images of semilinear and field-definable groups, Selecta Mathematica, vol. 18 (2012), no. 4, pp. 905–940.Google Scholar
[22] Eleftheriou, P. and Peterzil, Y., Definable quotients of locally definable groups, Selecta Mathematica, vol. 18 (2012), no. 4, pp. 885–903.Google Scholar
[23] Eleftheriou, P. and Peterzil, Y., Lattices in locally definable subgroups of 〈Rn, +〉, Notre Dame Journal of Formal Logic , to appear.Google Scholar
[24] Eleftheriou, P., Peterzil, Y., and Ramakrishnan, J., Interpretable groups are definable, preprint, 2011.Google Scholar
[25] Eleftheriou, P. and Starchenko, S., Groups definable in ordered vector spaces over ordered division rings, The Journal of Symbolic Logic, vol. 72 (2007), pp. 1108–1140.Google Scholar
[26] Günaydin, A. and Hieronymi, P., Dependent pairs, The Journal of Symbolic Logic, vol. 76 (2011), pp. 377–390.Google Scholar
[27] Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures, and the NIP, Journal of American Mathematical Society, vol. 21 (2008), pp. 563–596.Google Scholar
[28] Hrushovski, E., Peterzil, Y., and Pillay, A., On central extensions and definably compact groups in o-minimal structures, Journal of Algebra, vol. 327 (2011), pp. 71–106.Google Scholar
[29] Hrushovski, E. and Pillay, A.. On NIP and invariant measures, Journal of the European Mathematical Society, vol. 13 (2011), no. 4, pp. 1005–1061.Google Scholar
[30] Macpherson, D., Notes on o-minimality and variations, Model theory, algebra, and geometry (Haskell, D., Pillay, A., and Steinhorn, C., editors), Mathematical Sciences Research Institute Publications 39, Cambridge University Press, 2000, pp. 97–130.Google Scholar
[31] Marker, D., Peterzil, Y., and Pillay, A., Additive reducts of real closed fields, The Journal of Symbolic Logic, vol. 57 (1992), pp. 109–117.Google Scholar
[32] Miller, C. and Speissegger, P., Expansions of the real line by open sets: o-minimality and open cores, Fundamenta Mathematicae, vol. 162 (1999), no. 3, pp. 193–208.Google Scholar
[33] Otero, M., A survey on groups definable in o-minimal structures, Model theory with applications to algebra and analysis, Vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, 2008, pp. 177–206.Google Scholar
[34] Peterzil, Y., A structure theorem for semibounded sets in the reals, The Journal of Symbolic Logic, vol. 57 (1992), pp. 779–794.Google Scholar
[35] Peterzil, Y., Reducts of some structures over the reals, The Journal of Symbolic Logic, vol. 58 (1993), pp. 955–966.CrossRefGoogle Scholar
[36] Peterzil, Y., Returning io semi-bounded sets, The Journal of Symbolic Logic, vol. 74 (2009), pp. 597–617.Google Scholar
[37] Peterzil, Y. and Starchenko, S., A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society, vol. 77 (1998), no. 3, pp. 481–523.Google Scholar
[38] Peterzil, Y. and Starchenko, S., Definable homomorphisms of abelian groups in o-minimal structures, Annals of Pure and Applied Logic, vol. 101 (2000), no. 1, pp. 1–27.Google Scholar
[39] Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society, vol. 69 (1999), no. 2, pp. 769–786.Google Scholar
[40] Pillay, A., On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.Google Scholar
[41] Pillay, A., Scowcroft, P., and Steinhorn, C., Between groups and rings, Rocky Mountain Journal of Mathematics, vol. 19 (1989), no. 3, pp. 871–885.Google Scholar
[42] Strzebonski, A., Euler charateristic in semialgebraic and other o-minimal groups, Journal of Pure and Applied Algebra, vol. 96 (1994), pp. 173–201.Google Scholar