On some properties of base-matroids

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