拟阵的一个平集若为若干回路的并，则称为循环平集。拟阵的循环平集在包含关系下构成一个格。我们研究这些格，并从循环平集的视角考察拟阵。特别地，我们证明每个格都同构于某个拟阵的循环平集格。对于一个集合格 Z 以及 Z 上的函数 r，我们给出其成为某个拟阵的循环平集格及相应秩函数在 Z 上的限制的充要条件。我们定义了循环宽度，并证明该概念产生若干在取子式和对偶下封闭的拟阵类，其中两类仅包含横截拟阵。
英文摘要：
A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r on Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.
In this paper, first steps are taken towards characterising lattices of cyclic flats $\mathcal{Z}(M)$ that belong to matroids $M$ that can be represented over a prescribed finite field $\mathbb{F}_q$. Two natural maps from $\mathcal{Z}(M)$ to the lattice of cyclic flats of a minor of $M$ are given. Binary matroids are characterised via their lattice of cyclic flats. It is shown that the lattice of cyclic flats of a simple binary matroid without isthmuses is atomic.
We provide evidence for five long-standing, basis-exchange conjectures for matroids by proving them for the enormous class of sparse paving matroids. We also explore the role that these matroids may play in the following problem: as a function of the size of the ground set, what is the greatest number of cyclic flats that a matroid can have?
A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such as graphs, matrices, codes and projective geometries. In this work, we define cyclic matroids as matroids over a ground set of size $n$ whose automorphism group contains an $n$-cycle. We study the properties of such matroids, with special focus on the minimum size of their basis sets. For this, we broadly employ two different approaches: the multiple basis …
For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$ elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $σ$ of its ground set such that every $s-1$ consecutive elements of $σ$ are contained in an $s$-element circuit and every $t-1$ consecutive elements of $σ$ are contained in a $t$-element cocircuit. In the case $s=t$, nearly $(s, s)$-cyclic matroids have been studied previously. In this paper, we show that if $M$ is nearly $(s, t)$-cyclic and $n$ is sufficiently large, then these $s$-element circuits and $t$-element cocircuits are consecutive in $σ$ in…
We prove a new exchange property for bases of a matroid that generalizes the multiple symmetric exchange property. For every bases $B_1,\dots,B_k$ of a matroid and a subset $A_1\subset B_1$ there exist subsets $A_2\subset B_2,\dots,A_k\subset B_k$ such that all sets $(B_i\setminus A_i)\cup A_{i-1}$ achieved by a cyclic shift of $A_i$'s by one are bases.
We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid $M$, weight function $ω:E(M)\rightarrow\mathbb{N}$, and positive integer $D$, the following are equivalent. (1) For all $A\subseteq E(M)$, we have $\sum_{a\in A}ω(a)\le D\cdot r(A)$. (2) There is a map $ϕ$ that assigns to each element $e$ of $E(M)$ a set $ϕ(e)$ of $ω(e)$ cyclically consecutive elements in the cycle $(1,2,...,D)$ so that each set $\{e|i\inϕ(e)\}$, for $i=1,...,D$, is independent. As a first corollary we obtain the following. For each matroid $M$ so that $|E(M)|$ and $r(M)$ are c…
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 a similiar statement for perfect matroid designs due to Mphako and help to understand families of matroids with identical Tutte polynomial as constructed by Ken Shoda.
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $σ$ such that every set of $t-1$ consecutive elements in $σ$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $σ$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element cir…
We study the direct sum of q-matroids by way of their cyclic flats. Using that the rank function of a q-matroid is fully determined by the cyclic flats and their ranks, we show that the cyclic flats of the direct sum of two q-matroids are exactly all the direct sums of the cyclic flats of the two summands. This simplifies the rank function of the direct sum significantly. A q-matroid is called irreducible if it cannot be written as a (non-trivial) direct sum. We provide a characterization of irreducibility in terms of the cyclic flats and show that every q-matroid can be decomposed into a dire…
In this paper we develop the theory of cyclic flats of $q$-matroids. We show that the lattice of cyclic flats, together with their ranks, uniquely determines a $q$-matroid and hence derive a new $q$-cryptomorphism. We introduce the notion of $\mathbb{F}_{q^m}$-independence of an $\mathbb{F}_q$-subspace of $\mathbb{F}_q^n$ and we show that $q$-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.
We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined as the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$. We also show for $r \ge 2t$ that $M_{r,t}$ is the unique example for which equality holds.
We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$ projective geometry over GF$(q)$, where $q$ is the largest prime power not exceeding $\ell$.
Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic flats. We confirm a conjecture of Joswig and Schröter by proving an excluded-minor characterisation of the class of split matroids.
We present an output-sensitive algorithm for generating the whole set of flats of a finite matroid. Given a procedure, P, that decides in S_P time steps if a set is independent, the time complexity of the algorithm is O(N^2 M S_P), where N and M are the input and output size, respectively. In the case of vectorial matroids, a specific algorithm is reported whose time complexity is equal to O(N^2 M d^2), d being the rank of the matroid. In some cases this algorithm can provide an efficient method for computing zonotopes in $H$-representation, given their representation in terms of Minkowski sum…
We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional root system $H_4$. This root system coincides with the vertices of the 600-cell, a 4-dimensional regular solid. We determine the automorphism group of this matroid, showing half of the 14,400 automorphisms are geometric and half are not. We prove this group is transitive on the flats of the matroid, and also prove this group action is primitive. We use the incidence properties of the flats and the {\it orthoframes} of the matroid as a tool to understand these automorphisms, and interpret the flats geometrically.
Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats form a chain under inclusion; $M$ is laminar if and only if, for every $1$-element independent set $X$, the Hamiltonian flats of $M$ containing $X$ form a chain under inclusion. We generalize these notions to define the classes of $k$-closure-laminar and $k$-laminar matroids. This paper focuses on structural properties of these classes noting that, while the s…
A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroid context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of …
A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. …
A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most $2$.
Let $M$ be a matroid satisfying a matroidal analogue of the Cayley-Bacharach condition. Given a number $k \ge 2$, we show that there is no nontrivial bound on ranks of a $k$-tuple of flats covering the underlying set of $M$. This addresses a question of Levinson-Ullery motivated by earlier results which show that bounding the number of points satisfying the Cayley-Bacharach condition forces them to lie on low-dimensional linear subspaces. We also explore the general question what matroids satisfy the matroidal Cayley-Bacharach condition of a given degree and its relation to the geometry of gen…
The class of cographs or complement-reducible graphs is the class of graphs that can be generated from $K_1$ using the operations of disjoint union and complementation. By analogy, this paper introduces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary non-comatroids for which every proper flat is a binary comatroid. The paper also proves the…
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 this issue by developing theory towards code existence and design over a given field. This is done via exploiting recently established connections between linear locally repairable codes and matroids, and …
Motivated by the characterization of the lattice of cyclic flats of a matroid, the convolution of a ranked lattice and a discrete measure is defined, generalizing polymatroid convolution. Using the convolution technique we prove that if a matroid has a non-principal modular cut then it is not sticky. A similar statement for matroids has been proved in [8] using different technique.
Inspired by the work of Izakhian and Rhodes, a theory of representation of hereditary collections by boolean matrices is developed. This corresponds to representation by finite $\vee$-generated lattices. The lattice of flats, defined for hereditary collections, lattices and matrices, plays a central role in the theory. The representations constitute a lattice and the minimal and strictly join irreducible elements are studied, as well as various closure operators.
In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity to construct a convex geometry associated with an ordered, unoriented matroid. The construction in particular yields a new type of representability for an ordered matroid defined by the affine representability of its corresponding convex geometry. The lattice of convex sets of this convex geometry induces an ordering on the matroid independent sets which ex…
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman c…
