Linearly dependent vectorial decomposition of clutters

Abstract

This paper deals with the question of completing a monotone increasing family of subsets of a finite set to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we demonstrate that such vectorial completions of the family of subsets ¿ exist and, in addition, we show that the minimal vectorial completions of the family ¿ provide a decomposition of the clutter of the inclusion-minimal elements of ¿. The computation of such vectorial decomposition of clutters is also discussed in some cases.Peer ReviewedPostprint (author’s final draft