Matrix algebra of sets and variants of decomposition complexity

TL;DR

This paper develops a matrix algebra framework for sets in metric spaces to enhance understanding of asymptotic properties like asymptotic dimension, providing new characterizations and tools for decomposition complexity analysis.

Contribution

It introduces a novel matrix algebra approach for sets in metric spaces, improving results related to Asymptotic Property C and asymptotic dimension.

Findings

Characterization of asymptotic dimension using matrix algebra

Enhanced criteria for decomposition complexity

New tools for analyzing metric space properties

Abstract

We introduce matrix algebra of subsets in metric spaces and we apply it to improve results of Yamauchi and Davila regarding Asymptotic Property C. Here is a representative result: Suppose $X$ is an $\infty$-pseudo-metric space and $n\ge 0$ is an integer. The asymptotic dimension of $X$ is at most $n$ if and only if for any real number $r > 0$ and any integer $m\ge 1$ there is an augmented $m\times (n+1)$-matrix $\mathcal{M}=[\mathcal{B} |\mathcal{A}]$ (that means $\mathcal{B}$ is a column-matrix and $\mathcal{A}$ is an $m\times n$-matrix) of subspaces of $X$ of scale-$r$-dimension $0$ such that $\mathcal{M}\cdot_\cap \mathcal{M}^T$ is bigger than or equal to the identity matrix and $B(\mathcal{A},r)\cdot_\cap B(\mathcal{A},r)^T$ is a diagonal matrix.