Matrix algebra of sets and variants of decomposition complexity

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 $\mathrm{\infty }$-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 $\mathcal{M}=[\mathcal{B}\mathrm{\mid }\mathcal{A}]$ (that means $\mathcal{B}$ is a column-matrix and $\mathcal{A}$ is an $m\times n$-matrix)ofsubspacesof$X$ofscale-$r$-dimension$0$suchthat$$\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.