Related papers: Semidirect sums of matroids

We develop a theory of measures, differential forms and Fourier tramsforms on some infinite-dimensional real vector spaces by generalizing the following two constructions: (a) The construction of the semiinfinite wedge power of a Tate…

Gross substitutability is a central concept in Economics and is connected to important notions in Discrete Convex Analysis, Number Theory and the analysis of Greedy algorithms in Computer Science. Many different characterizations are known…

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that $S(k,\ell) = \mbox{tr}(A_k^*…

We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a…

Let A be a singular matrix of M_n(K), where K is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of (M_n(K),x) generated by the similarity class of A is the set of matrices of M_n(K) with a rank lesser…

The direct product $\mathbb{N}\times\mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise non-isomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite…

It is shown that the category of semi-biproducts in monoids is equivalent to a category of pseudo-actions. A semi-biproduct in monoids is at the same time a generalization of a semi-direct product in groups and a biproduct in commutative…

The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits,…

In this paper which is the first of a series of papers on smooth structures, the concepts of C-structures and smooth structures are introduced and studied. The notion of smooth structure on semi-integral domains is given. It is shown that…

For ${n \in \mathbb{N}}$, the $n$-truncation of a matroid $M$ of rank at least $n$ is the matroid whose bases are the $n$-element independent sets of $M$. One can extend this definition to negative integers by letting the $(-n)$-truncation…

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a…

q-Matroids form the q-analogue of classical matroids. In this paper we introduce various types of maps between q-matroids. These maps are not necessarily linear, but they map subspaces to subspaces and respect the q-matroid structure in…

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…

A notion of {\em normal submonoid} of a monoid $M$ is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf{NorSub}(M)$ of normal submonoids of $M$ is a complete lattice. Joins are…

A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability…

A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always…

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via…