The Category of Matroids

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial.Comment: 31 pages, 10 diagrams, 28 reference

COMPATIBILITY OF EXTENSIONS OF A COMBINATORIAL GEOMETRY

Two extensions of a geometry are compatible with each other if they have a common extension. If the given extensions are elementary, their compatibility can be intrinsically described in terms of their corresponding linear subclasses. Certain adjointness relation between an extension of a geometry and the geometry itself is also discussed. Any extension of a geometry G by a geometry F determines and is determined by a unique quotient bundle on G indexed by F. As a study of the compatibility among given quotients of a geometry, we look at the possibility of completing to F-bundles a family of quotients indexed by a set I of flats of F. If the indexing geometry F is free and if the set I is a Boolean subalgebra or a sublattice of F, for any family Q(I) of quotients of a geometry G, there is a canonical construction which determines its completability and at the same time produces the extremal completion if it is a partial bundle. Geometries studied in this dissertation are furnished with the weak order. Almost invariably, the Higgs' lift construction, in a somewhat generalized sense, constitutes a convenient and indispensable means in various of the extremal constructions

Theory and applications of freedom in matroids

To each cell e in a matroid M we can associate a non-negative integer ǁ e ǁ called the freedom of e. Geometrically the value ǁ e ǁ indicates how freely placed the cell is in the matroid. We see that ǁ e ǁ is equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e. The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and the applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of M*. If ζ(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M ǁ e ǁ= \frac{max\ f \ (e)}{f\in\zeta} We look at freedom in binary matroids and show that for a connected binary matroid M, ǁ e ǁ is the number of connected components of M/e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M/e disconnected