Related papers: Sticky matroids and convolution
We prove that any element in a matroid can be removed, by either deletion or contraction, in such a way that no tangle "splits".
We give a criterion for modular extension of rank-4 hypermodular matroids, and prove a weakening of Kantor's conjecture for rank-4 realizable matroids. This proves the sticky matroid conjecture and Kantor's conjecture for realizable…
Cyclic flats form a common structural invariant of both matroids and $q$-matroids, determining these objects through their weighted lattices of cyclic flats. In this paper we exploit this perspective to establish a correspondence between…
Matroid varieties are the closures in the Grassmannian of sets of points defined by specifying which Pl\"ucker coordinates vanish and which don't --- the set of nonvanishing Pl\"ucker coordinates forms a well-studied object called a…
We generalize the well-studied notion of a modular pair of a finite matroid to arbitrary families of sets in infinite matroids, and use it to develop the theory of infinite matroids in several as-yet-unexplored areas. Our results include a…
It is unknown so far, whether the lattice of all varieties of monoids satisfies some non-trivial identity. The objective of this note is to give the negative answer to this question. Namely, we prove that any finite lattice is a homomorphic…
Adhesive polymatroids were defined by F. Mat\'u\v{s} motivated by entropy functions. Two polymatroids are adhesive if they can be glued together along their joint part in a modular way; and are one-adhesive, if one of them has a single…
A ring is rigid if there is no nonzero locally nilpotent derivation on it. In terms of algebraic geometry, a rigid coordinate ring corresponds to an algebraic affine variety which does not allow any nontrivial algebraic additive group…
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…
A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $\mathrm{PG}(n-1,2)$ for which $|E \cap P|$ is not a basis of $P$ for any…
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…
It is well known that the class of transversal matroids is not closed under contraction or duality. In particular, after contracting a set of elements from a transversal matroid, the resulting matroid may or may not be transversal, and the…
We consider structural properties of $\mathcal{L}$-polymatroids, especially those defined on a finite complemented modular lattice $\mathcal{L}$. We introduce a set of cover-weight axioms and establish a cryptomorphism between these axioms…
Matrix stiffness expressions are derived for the particle movements in an assembly of rigid granules having compliant contacts. The derivations include stiffness terms that arise from the particle shapes at their contacts. These geometric…
We study a matrix-based notion of matroid representation over local commutative rings obtained by replacing linear independence with modular independence. This construction always defines an independence system, though not necessarily a…
The sticking of a soft polystyrene colloidal particle to a planar glass plate was studied by a microrheological technique using an optical tweezer to trap the particle and a piezoelectric-stage to position the plate and to sinusoidally…
We show that there exist $k$-colorable matroids that are not $(b,c)$-decomposable when $b$ and $c$ are constants. A matroid is $(b,c)$-decomposable, if its ground set of elements can be partitioned into sets $X_1, X_2, \ldots, X_l$ with the…
In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a…