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arXiv · 1612.06771

Matrix algebra of sets and variants of decomposition complexity

Jerzy Dydak
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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.

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Jerzy Dydak. 2016-12-20. Matrix algebra of sets and variants of decomposition complexity. https://arxiv.org/abs/1612.06771

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