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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Deza, M., Laurent, M. Bouquets of matroids, d-injection geometries and diagrams. J Geom 29, 12–35 (1987). https://doi.org/10.1007/BF01234984
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DOI: https://doi.org/10.1007/BF01234984