Bouquets of matroids,d-injection geometries and diagrams |
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Authors: | Michel Deza Monique Laurent |
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Institution: | (1) CNRS, 17 Passage de l'Industrie, 750 Paris, France;(2) CNET PAA-TIM, 38-40 rue du Général Leclerc, 92 131, Issy Les Moulineaux, France |
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Abstract: | F-squashed geometries, one of the many recent generalizations of matroids, include a wide range of combinatorial structures but still admit a direct extension of many matroidal axiomatizations and also provide a good framework for studying the performance of the greedy algorithm in any independence system. Here, after giving all necessary preliminaries in section 1, we consider in section 2F-squashed geometries which are exactly the shadow structures coming from the Buekenhout diagram:, i.e. bouquets of matroids. We introduce d-injective planes: (generalizing the case of dual net for d=1) which provide a diagram representation for high rank d-injective geometries. In section 3, after a brief survey of known constructions for d-injective geometries, we give two new constructions using pointwise and setwise action of a class of mappings. The first one, using some features of permutation geometries (i.e. 2-injection geometries), produces bouquets of pairwise isomorphic matroids. The last section 4 presents briefly some related problems for squashed geometries. |
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