Definable groups in topological differential fields

TL;DR

This paper explores the structure of definable groups in topological differential fields, establishing connections between definability in extended languages and reducts, and demonstrating properties like the descending chain condition in specific models.

Contribution

It introduces a method to relate definable sets in enriched languages to their reducts and constructs local definable groups from global definable groups in topological differential fields.

Findings

Relationship between definable sets in extended and reduct languages

Construction of local definable groups from global groups

Descending chain condition on centralisers in closed ordered differential fields

Abstract

For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between $\mathcal L\cup\{D\}$-definable sets and their $\mathcal L$-reducts, where $\mathcal L$ is a relational expansion of the field language and $D$ a symbol for a derivation. This enables us to associate with an $\mathcal L\cup\{D\}$-definable group in models of such theories, a local $\mathcal L$-definable group. As a byproduct, we show that in closed ordered differential fields, one has the descending chain condition on centralisers.