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The main result is Theorem MAT 11 which states that every finite closureoperator is the ground set of a matroid. Its base sets consist of nonredundantcovers of of the closure. These are minimal subsets that determine the closureoperator using a closure algorithm from the theory of relational databases. Foreach hereditary collection there is one flat closure that define a matroid orsimplicial complexes, but every closure defines a hereditary collection, itskeys in database theory. The normalization algorithm by Maier is the basis ofthis paper. This follows his development, and his relevant results are cited toaid reading his original work. The main purpose of this paper is to introducethe importance of database theory into the analysis of all finite closureincluding the flat closures of matroids.

A Relational Approach to Matroids, Simplicial Complexes and Finite Closures