Related papers: Representing matroids via pasture morphisms

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave…

We generalise the construction of infinite matroids from trees of matroids to allow the matroids at the nodes, as well as the field over which they are represented, to be infinite.

Rough sets were proposed to deal with the vagueness and incompleteness of knowledge in information systems. There are may optimization issues in this field such as attribute reduction. Matroids generalized from matrices are widely used in…

We give an overview of the Macaulay2 package Matroids, which contains functionality to create and compute with matroids. Examples highlighting the use of all major functions in the package are provided, along with explanations of some of…

We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R=$\mathbb{Z}$, and when R is a DVR, we get…

We study realization spaces of matroids over hyperfields (in the sense of Baker and Bowler). More precisely, given a matroid M and a hyperfield H we determine the space of all H-matroids over M. This can be seen as the matroid stratum of…

Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…

A matroid is $\text{GF}(q)$-regular if it is representable over all proper superfields of the field $\text{GF}(q)$. We show that, for highly connected matroids having a large projective geometry over $\text{GF}(q)$ as a minor, the property…

The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate, the natural matroid can shed much more light on integer…

The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of ``sparse'' paving matroids that are balanced, but at the same time rich…

Attribute reduction is a basic issue in knowledge representation and data mining. Rough sets provide a theoretical foundation for the issue. Matroids generalized from matrices have been widely used in many fields, particularly greedy…

In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a variant of Viro's patchworking to derive a topological…

This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…

Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence…

We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary…

We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this…

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex…

Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and…