On Unary Functions Definable in Coset-Minimal Ordered Groups

Viktor V. Verbovskiy

A totally ordered group G is coset-minimal [1] if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G ∪ {−∞, +∞}. We study definable unary functions in coset-minimal ordered groups. Main results: Every unary definable function in a discrete coset-minimal ordered group is piecewise linear. Every unary definable function in a dense cosetminimal ordered group is piecewise monotone. Let G be dense and F the family of all definable unary functions f : G_{1} → G_{1}, where G_{1} ≤ G is convex definable and o-minimal and G/G_{1} is discrete. Then any G-definable unary function is definable in the reduct to the language {<, +, F}. Dense coset-minimal ordered groups have the Exchange Property.

Definition 1 [1] Let f be a definable function from a group G to a group H. Then f is piecewise linear if there is a finite partition of G into definable sets {X_{i} : i ∈ I}, elements g_{i} ∈ G and h_{i} ∈ H and homomorphisms σ_{i} : ⟨X_{i} − g_{i}⟩ → H for all i ∈ I, such that f(x + g_{i}) − h_{i} is the restriction of σ_{i} to X_{i} − g_{i} for all i ∈ I.

[1] F. Point and F. Wagner, Essentially periodic ordered groups, to appear in: Communications in Algebra.

Address: Institute of Control and Informatics Problems, Ministry of Education and Science, 480100, ul. Pushkina 125, Almaty, Kazakhstan

E-mail: vvver@mail.ru