Some Properties of Cyclic Flats of an Infinite Matroid
نویسنده
چکیده
We prove that when a pre-independence space satisfies some natural properties, then its cyclic flats form a bounded lattice under set inclusion. Additionally, we show that a bounded lattice is isomorphic to the lattice of cyclic flats of a pre-independence space. We also prove that the notion of cyclic width gives rise to dual-closed and minorclosed classes of B-matroids. Finally, we find a difference between finite matroids and B-matroids by using the notion of well-quasi-ordering.
منابع مشابه
The Lattice of Cyclic Flats of a Matroid
Matroid theory is a combinatorial abstraction of geometry, with flats playing the role of subspaces. Cyclic flats are special flats that contain key geometric information about a matroid. This talk presents a variety of recent results and open problems about the lattice of cyclic flats. In particular, we show that every finite lattice arises as the lattice of cyclic flats of a (fundamental tran...
متن کاملOn Binary Matroid Minors and Applications to Data Storage over Small Fields
Locally repairable codes for distributed storage systems have gained a lot of interest recently, and various constructions can be found in the literature. However, most of the constructions result in either large field sizes and hence too high computational complexity for practical implementation, or in low rates translating into waste of the available storage space. In this paper we address th...
متن کاملComputing the Tutte Polynomial of a Matroid from its Lattice of Cyclic Flats
We show how the Tutte polynomial of a matroid M can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of M is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks. They furthermore generalize similiar statements for perfect matroid designs and near d...
متن کاملThe Higgs Factorization of a Geometric Strong
The Higgs faotopization of a strong map between matroids on a fixed set is that factorization into elementary naps in which each matroid is the Higgs lift of its successor. This factorization is characterized by properties of the modular filters which induce the elementary maps of the factorizations in two different ways. It is also sho~m to be minimal in a natural order on factorizations arisi...
متن کاملTransversal Lattices
A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats are isomorphic to it are transversal. We investigate some sufficient conditions for a lattice to be a Tr-lattice; a corollary is that distributive lattices of ...
متن کامل