203 research outputs found
Foundations of matroids -- Part 2: Further theory, examples, and computational methods
In this sequel to "Foundations of matroids - Part 1", we establish several
presentations of the foundation of a matroid in terms of small building blocks.
For example, we show that the foundation of a matroid M is the colimit of the
foundations of all embedded minors of M isomorphic to one of the matroids
, , , , , , ,
, and we show that this list is minimal. We establish similar minimal
lists of building blocks for the classes of 2-connected and 3-connected
matroids. We also establish a presentation for the foundation of a matroid in
terms of its lattice of flats. Each of these presentations provides a useful
method to compute the foundation of certain matroids, as we illustrate with a
number of concrete examples. Combining these techniques with other results in
the literature, we are able to compute the foundations of several interesting
classes of matroids, including whirls, rank-2 uniform matroids, and projective
geometries. In an appendix, we catalogue various 'small' pastures which occur
as foundations of matroids, most of which were found with the assistance of a
computer, and we discuss some of their interesting properties.Comment: 69 page
Cubes and orientability
AbstractWe define and study a new class of matroids: cubic matroids. Cubic matroids include, as a particular case, all affine cubes over an arbitrary field. There is only one known orientable cubic matroid: the real affine cube. The main results establish as an invariant of orientable cubic matroids the structure of the subset of acyclic orientations with LV-face lattice isomorphic to the face lattice of the real cube or, equivalently, with the same signed circuits of length 4 as the real cube
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