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\(\mathrm {asdim}(X)\) 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.

中文翻译：

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集和分解复杂度的矩阵代数

我们引入度量空间中子集的矩阵代数，并将其用于改进Yamauchi和Davila关于渐近性质C的结果。这是一个代表性的结果：假设X是\（\ infty \）-伪度量空间和\（n \ ge 0 \）是整数。该渐近尺寸\（\ mathrm {asdim}（X）\）的X是至多Ñ当且仅当对于任何实数\（R> 0 \）和任何整数\（米\ GE 1 \）有一个扩充\（m \，\ times \，（n + 1）\）- matrix \（{\ mathcal {M}} = [{\ mathcal {B}} | {\ mathcal {A}}] \）（）表示\（{\ mathcal {B}} \）是一个列矩阵和\（{\ mathcal {A}} \）是\（米\，\倍\，正\）的子空间的-矩阵）X横向扩展的- [R -尺寸0，使得\（{ \ mathcal {M}} \ cdot _ \ cap {\ mathcal {M}} ^ T \）大于或等于恒等矩阵和\（B（{\ mathcal {A}}，r）\ cdot _ \ cap B（{\ mathcal {A}}，r）^ T \）是对角矩阵。