Coset-minimal groups

Abstract

A totally ordered group G (possibly with extra structure) 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 boolean OR {+/-infinity}. Continuing work in Belegradek et al. (J. Symbolic Logic 65(3) (2000) 1115) and Point and Wagner (Ann. Pure Appl. Logic 105(1-3) (2000) 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 (this improves a result in Point and Wagner (Ann. Pure Appl. Logic 105(1-3) (2000) 261), where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property (which is false in the discrete case (M.S.R.I., preprint series, 1998-051)); 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. (C) 2002 Elsevier Science B.V. All rights reserved.