On some properties of base-matroids |
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Authors: | F Maffioli |
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Institution: | a Dip. di Elettronica ed Informazione, Politecnico di Milano, P.zza L. da Vinci, 32, 20133 Milano, Italy b Dip. di Matematica, Politecnico di Milano, P.zza L. da Vinci 32, 20133 Milano, Italy |
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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. |
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Keywords: | Matroid Base Uniform matroid Lattice Boolean lattice |
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