Related papers: Matroid intersection, base packing and base coveri…
We prove that if $ M $ and $ N $ are finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $…
In this note we characterize tropical bases as sets of circuits that by orthogonality determine the set of cocircuits of a simple matroid. Furthermore, we show that any circuit, which itself is closed, must be contained in any tropical…
We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, ${\mathcal T}_{\rm min}(E)$ and ${\mathcal T}_{\rm ort}(E)$. We show that these three configurations are…
We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and…
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
We prove that the set of permutations sorted by a stack of depth $t \geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a…
Recently, it was proved by B\'erczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were…
In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids. In…
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…
The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation".
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…
We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We…
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…
We review a few results concerning interpolation of monotone functions on infinite lattices, emphasizing the role of set-theoretic considerations. We also discuss a few open problems.
Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the…
We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for…
The discrete polymatroids and their base rings are studied recently in many papers (see \cite{HH}, \cite{HHV}, \cite{V1}, \cite{V2}). It is important to give conditions when the base ring associated to a transversal polymatroid is…
In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our…