167 research outputs found
Matroids with nine elements
We describe the computation of a catalogue containing all matroids with up to
nine elements, and present some fundamental data arising from this cataogue.
Our computation confirms and extends the results obtained in the 1960s by
Blackburn, Crapo and Higgs. The matroids and associated data are stored in an
online database, and we give three short examples of the use of this database.Comment: 22 page
A module-theoretic approach to matroids
Speyer recognized that matroids encode the same data as a special class of
tropical linear spaces and Shaw interpreted tropically certain basic matroid
constructions; additionally, Frenk developed the perspective of tropical linear
spaces as modules over an idempotent semifield. All together, this provides
bridges between the combinatorics of matroids, the algebra of idempotent
modules, and the geometry of tropical linear spaces. The goal of this paper is
to strengthen and expand these bridges by systematically developing the
idempotent module theory of matroids. Applications include a geometric
interpretation of strong matroid maps and the factorization theorem; a
generalized notion of strong matroid maps, via an embedding of the category of
matroids into a category of module homomorphisms; a monotonicity property for
the stable sum and stable intersection of tropical linear spaces; a novel
perspective of fundamental transversal matroids; and a tropical analogue of
reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA
On ternary transversal matroids
AbstractThe purpose of this paper is to answer a question of Ingleton by characterizing the class of ternary transversal matroids
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