Related papers: Separations of sets
There has been a great deal of research on graphs defined on algebraic structures in the last two decades. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this…
A set $V$ is said to be separated by subsets $V_1,\ldots,V_k$ if, for every pair of distinct elements of $V$, there is a set $V_i$ that contains exactly one of them. Imposing structural constraints on the separating subsets is often…
We define an independence system associated with simple graphs. We prove that the independence system is a matroid for certain families of graphs, including trees, with bases as minimal resolving sets. Consequently, the greedy algorithm on…
At its core, abstraction is the process of generalizing from specific instances to broader concepts or models, with the primary objective of reducing complexity while preserving properties essential to the intended purpose. It is…
We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality…
Structural causal models (SCMs) are a widespread formalism to deal with causal systems. A recent direction of research has considered the problem of relating formally SCMs at different levels of abstraction, by defining maps between SCMs…
The continuous and rapid growth of highly interconnected datasets, which are both voluminous and complex, calls for the development of adequate processing and analytical techniques. One method for condensing and simplifying such datasets is…
A set of general physical principles is proposed as the structural basis for the theory of complex systems. First the concept of harmony is analyzed and its different aspects are uncovered. Then the concept of reflection is defined and…
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of…
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
The matching complex of a graph is the simplicial complex whose vertex set is the set of edges of the graph with a face for each independent set of edges. In this paper we completely characterize the pairs (graph, matching complex) for…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
Linear diagrams are used to visualize set systems by depicting set memberships as horizontal line segments in a matrix, where each set is represented as a row and each element as a column. Each such line segment of a set is shown in a…
Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our…
We describe a basic correspondence between linear algebraic structures within vector embeddings in artificial neural networks and conditional independence constraints on the probability distributions modeled by these networks. Our framework…
A common technique for checking properties of complex state machines is to build a finite abstraction then check the property on the abstract system -- where a passing check on the abstract system is only transferred to the original system…
Systems of fixpoint equations over complete lattices, consisting of (mixed) least and greatest fixpoint equations, allow one to express a number of verification tasks such as model-checking of various kinds of specification logics or the…
We describe the structure of triconnected graph with the help of its decomposition by 3-cutsets. We divide all 3-cutsets of a triconnected graph into rather small groups with a simple structure, named complexes. The detailed description of…
We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a…
The abstract hyperbolic sets are introduced. Continuous and differentiable mappings as well as rate of convergence and transversal manifolds are not under discussion, and the symbolic dynamics paradigm is realized in a new way. Our…