Linear dilatation structures and inverse semigroups

TL;DR

This paper establishes a new equivalence between linear dilatation structures and properties of inverse semigroups generated by dilatations, providing novel insights even for vector spaces and Carnot groups.

Contribution

It proves that linearity of dilatation structures is equivalent to a specific property of the inverse semigroup generated by dilatations, a result new for Carnot groups and vector spaces.

Findings

Linearity of dilatation structures relates to inverse semigroup properties.

The proof introduces a new perspective for understanding dilatation structures.

Results are novel for both Carnot groups and vector spaces.

Abstract

Here we prove that for dilatation structures linearity (see arXiv:0705.1440v1) is equivalent to a statement about the inverse semigroup generated by the family of dilatations of the space. The result is new for Carnot groups and the proof seems to be new even for vector spaces.