Preliminaries for Squashed Geometries and Diagrams A) Preliminaries for Squashed Geometries. Definition 1.1. Let X Be a Finite Set, F Be a Clutter of Subsets of X And

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 ], we consider in section 2 F-squashed geometries which are exactlYLthe s~adow L structures coming from the Buekenhout diagram : o a...o [qjT-----Oaj o , i.e. bouquets of matroids. We introduce d-injective planes : o (generalizing the case of dual net for d = I) 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.