Coset-minimal groups

Annals of Pure and Applied Logic 121 (2-3):113-143 (2003)

Abstract
A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. Any pure ordered abelian group of finite regular rank is ultimately coset-minimal and has the exchange property; moreover, every definable function in such a group is piecewise linear. Pure coset-minimal and eventually coset-minimal groups are classified. In a discrete coset-minimal group every definable unary function is piece-wise linear 261), where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property ); moreover, any definable unary function is piecewise linear, except possibly for finitely many cosets of the smallest definable convex nonzero subgroup. Finally, we give some examples and open questions
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DOI 10.1016/s0168-0072(02)00084-2
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References found in this work BETA

Quasi-o-Minimal Structures.Oleg Belegradek, Ya'acov Peterzil & Frank Wagner - 2000 - Journal of Symbolic Logic 65 (3):1115-1132.
Extended Order-Generic Queries.Oleg V. Belegradek, Alexei P. Stolboushkin & Michael A. Taitslin - 1999 - Annals of Pure and Applied Logic 97 (1-3):85-125.
Essentially Periodic Ordered Groups.Françoise Point & Frank O. Wagner - 2000 - Annals of Pure and Applied Logic 105 (1-3):261-291.

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Citations of this work BETA

Semi-Bounded Relations in Ordered Modules.Oleg Belegradek - 2004 - Journal of Symbolic Logic 69 (2):499 - 517.
Semi-Bounded Relations in Ordered Modules.Oleg Belegradek - 2004 - Journal of Symbolic Logic 69 (2):499-517.

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