TY - JOUR

T1 - Defining integer valued functions in rings of continuous definable functions over a topological field

AU - Darnière, Luck

AU - Tressl, Marcus

N1 - Publisher Copyright:
© 2020 World Scientific Publishing Company.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/1/30

Y1 - 2020/1/30

N2 - Let K be an expansion of either an ordered field or a valued field. Given a definable set X ⊆ Km let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f ∈ C(X) which take values in Z is definable in C(X).

AB - Let K be an expansion of either an ordered field or a valued field. Given a definable set X ⊆ Km let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f ∈ C(X) which take values in Z is definable in C(X).

KW - Rings of Continuous Functions, Defining Integers, Topological Fields, Definable Functions

UR - https://arxiv.org/abs/1810.12562

U2 - 10.1142/S0219061320500142

DO - 10.1142/S0219061320500142

M3 - Article

SN - 0219-0613

VL - 20

SP - 1

EP - 24

JO - Journal of Mathematical Logic

JF - Journal of Mathematical Logic

IS - 3

M1 - 2050014

ER -