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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 $$n\ge 0$$ n ≥ 0 is an integer. The asymptotic dimension $$\mathrm {asdim}(X)$$ asdim ( X ) of X is at most n if and only if for any real number $$r > 0$$ r > 0 and any integer $$m\ge 1$$ m ≥ 1 there is an augmented $$m\,\times \,(n+1)$$ m × ( n + 1 ) -matrix $${\mathcal {M}}=[{\mathcal {B}} |{\mathcal {A}}]$$ M = [ B | A ] (that means $${\mathcal {B}}$$ B is a column-matrix and $${\mathcal {A}}$$ A is an $$m\,\times \,n$$ m × n -matrix) of subspaces of X of scale- r -dimension 0 such that $${\mathcal {M}}\cdot _\cap {\mathcal {M}}^T$$ M · ∩ M T is bigger than or equal to the identity matrix and $$B({\mathcal {A}},r)\cdot _\cap B({\mathcal {A}},r)^T$$ B ( A , r ) · ∩ B ( A , r ) T is a diagonal matrix.

Matrix algebra of sets and variants of decomposition complexity