On some properties of base-matroids

  • Authors:

  • F. Maffioli;N. Zagaglia Salvi

  • Affiliations:

  • Dip. di Elettronica ed Informazione, Politecnico di Milano, Milano, Italy;Dip. di Matematica, Politecnico di Milano, Milano, Italy

  • Venue:
  • Discrete Applied Mathematics - Special issue: 2nd cologne/twente workshop on graphs and combinatorial optimization (CTW 2003)
  • Year:
  • 2006

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Abstract

Let M = (E, F) be a rank-n matroid on a set E and B one of its bases. A closed set θ ⊆ E is saturated with respect to B, or B-saturated, when |θ∩B| = r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I∩θ| ≤ r(θ), for every closed B-saturated set θ, turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by MB. In this paper we prove some properties of MB, in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to MB for every base B of M.Finally, we prove that the poset of the closed B-saturated sets ordered by inclusion is isomorphic to the Boolean lattice Bn.