Related papers: Matroids over Skew Tracts
We develop two generalizations of contraction theory, namely, semi-contraction and weak-contraction theory. First, using the notion of semi-norm, we propose a geometric framework for semi-contraction theory. We introduce matrix…
Using the framework of pastures and foundations of matroids developed by Baker-Lorscheid, we give algorithms to: (i) compute the foundation of a matroid, and (ii) compute all morphisms between two pastures. Together, these provide an…
We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups…
Projective metrics on vector spaces over finite fields, introduced by Gabidulin and Simonis in 1997, generalize classical metrics in coding theory like the Hamming metric, rank metric, and combinatorial metrics. While these specific metrics…
Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. Conversely, we…
We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of…
We extend the the combinatorics of tableaux to the study of diagram algebras and give a uniform construction of their quasi-hereditary covers.
Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…
In this article we make several contributions of independent interest. First, we introduce the notion of stressed hyperplane of a matroid, essentially a type of cyclic flat that permits to transition from a given matroid into another with…
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra…
In this paper we derive an extended Circle Pattern Theorem that allows obtuse overlap angles. As a consequence, we characterize a subclass of compact convex hyperbolic polyhedra with possibly obtuse dihedral angles and thus generalize…
Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint…
We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.
We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k=2) and…
We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of…
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…
In this paper, we establish a one-to-one correspondence between the set of biclosed sets in an irreducible root system of type $A_n$ and the set of quasitrivial semigroup structures on a set with $n+1$ elements. Building on this…
We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm…
Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented…
We study matroidal networks introduced by Dougherty et al. We prove the converse of the following theorem: If a network is scalar-linearly solvable over some finite field, then the network is a matroidal network associated with a…