Matrix Invariants as Homotopy Invariants in Finite $T_0$-spaces

Abstract: We establish a bijection between the set of finite topological $T_0$-spaces (or partially ordered sets) and classes of square matrices. The absolute value of the determinant or the rank of these matrices are simple homotopy invariants for the corresponding topological spaces, and consequently, for finite simplicial complexes. We calculate these invariants for various families of topological spaces and explore other relationships and problems posed for finite posets within the context of these matrices.