Essentially periodic ordered groups

✍ Scribed by Françoise Point; Frank O. Wagner

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 246 KB
• Volume: 105
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(99)00053-6

✦ Synopsis

A totally ordered group G is essentially periodic if for every deÿnable non-trivial convex subgroup H of G every deÿnable subset of G is equal to a ÿnite union of cosets of subgroups of G on some interval containing an end segment of H ; it is coset-minimal if all deÿnable subsets are equal to a ÿnite union of cosets, intersected with intervals. We study deÿnable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek et al. (J. Symbolic Logic, to appear). Main results: An essentially periodic group G is abelian; if G is discrete, then deÿnable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal (and vice versa), and its deÿnable functions in one variable are piecewise linear.