Linearly dependent vectorial decomposition of clutters

Abstract

This paper deals with the question of completing a monotone increas- ing family of subsets G of a finite set ¿ to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we demonstrate that such vec- torial completions of the family of subsets G exist and, in addition, we show that the minimal vectorial completions of the family G provide a decomposition of the clutter ¿ of the inclusion-minimal elements of G . The computation of such vectorial decomposition of clutters is also discussed in some cases