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Weakly one-based geometric theories. (English) Zbl 1405.03077

Summary: We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based \(\omega \)-categorical geometric theory interprets an infinite vector space over a finite field.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C64 Model theory of ordered structures; o-minimality
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References:

[1] DOI: 10.1016/0168-0072(84)90004-6 · Zbl 0588.03014
[2] DOI: 10.1016/j.apal.2007.06.002 · Zbl 1171.03020
[3] Proceedings of the London Mathematical Society pp 481– (2000)
[4] DOI: 10.1016/0022-4049(94)90007-8 · Zbl 0832.03019
[5] Construction d’un groupe dans les structures C-minimales 73 pp 957– (2008) · Zbl 1162.03020
[6] DOI: 10.1007/BF02761295 · Zbl 0797.03034
[7] A first course in non-commmutative rings (2001)
[8] DOI: 10.1007/s11856-010-0085-y · Zbl 1213.03050
[9] The theory of groups (1959)
[10] DOI: 10.1016/j.apal.2004.10.016 · Zbl 1064.03024
[11] Characterizing rosy theories 72 pp 919– (2007)
[12] DOI: 10.1016/S0168-0072(03)00018-6 · Zbl 1030.03026
[13] Simple theories (2000)
[14] DOI: 10.1016/S0168-0072(02)00060-X · Zbl 1010.03023
[15] DOI: 10.1017/S0024611506015747 · Zbl 1101.03028
[16] Fundamenta Mathematicae 157 pp 61– (1998)
[17] DOI: 10.1305/ndjfl/1134397664 · Zbl 1097.03025
[18] Geometric stability theory (1996)
[19] DOI: 10.1090/S0002-9947-03-03327-0 · Zbl 1021.03023
[20] Annals of Pure and Applied Logic pp 71– (1998)
[21] Essential stability theory 4 (1996) · Zbl 0864.03025
[22] DOI: 10.1016/j.apal.2009.10.004 · Zbl 1227.03044
[23] Paires de structures o-minimales 63 pp 570– (1998)
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