A note on clustered cells

TL;DR

This paper discusses improvements and clarifications related to clustered cell decomposition in P-minimal structures, focusing on the order of clustered cells and introducing a notion of minimal order cells.

Contribution

It provides additional insights and a refined concept of minimal order cells, building upon previous work on clustered cell decomposition in P-minimal structures.

Findings

Clarifies the order of clustered cells

Introduces a notion of cells of minimal order

Provides refinements to existing theorems

Abstract

This note contains additions to the paper 'Clustered cell decomposition in P-minimal structures' (arXiv:1612.02683). We discuss a question which was raised in that paper, on the order of clustered cells. We also consider a notion of cells of minimal order, which is a slight optimalisation of the theorem from the original paper.