Related papers: Matroids, hereditary collections and simplicial co…
We introduce a new representation concept for lattices by boolean matrices, and utilize it to prove that any matroid is boolean representable. We show that such a representation can be easily extracted from a representation of the…
We extend the notion of matroid representations by matrices over fields and consider new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of…
It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic…
Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of…
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
In this paper we study the representation theory of the algebras generated by the single bond transfer matrices in dilute lattice models. This representation theory is related to a tensor product of monoidal categories. This construction is…
The main result is Theorem MAT 11 which states that every finite closure operator is the ground set of a matroid. Its base sets consist of nonredundant covers of of the closure. These are minimal subsets that determine the closure operator…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…
We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
We determine all composition-closed equational classes of Boolean functions. These classes provide a natural generalization of clones and iterative algebras: they are closed under composition, permutation and identification…
We call a class $\mathcal{M}$ of matroids hereditary if it is closed under flats. We denote by $\mathcal{M}^{ext}$ the class of matroids $M$ that is in $\mathcal{M}$, or has an element $e$ such that $M \backslash e$ is in $\mathcal{M}$. We…
Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through…
We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a…
Efficient deterministic algorithms to construct representations of lattice path matroids over finite fields are presented. They are built on known constructions of hierarchical secret sharing schemes, a recent characterization of…
We examine the lattice generated by two pairs of supplementary vector subspaces of a finite-dimensional vector space by intersection and sum, with the aim of applying the results to the study of representations admitting two pairs of…
Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and…
We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which…
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the…
Birkhoff's representation theorem (Birkhoff, 1937) defines a bijection between elements of a distributive lattice and the family of upper sets of an associated poset. Although not used explicitly, this result is at the backbone of the…