Related papers: Opetopes and chain complexes
Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and…
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…
A network-theoretic approach for determining the complexity of a graph is proposed. This approach is based on the relationship between the linear algebra (theory of determinants) and the graph theory. In this paper we contribute a new…
Here we introduce simple structures for the analysis of complex hypergraphs, hypergraph animals. These structures are designed to describe the local node neighbourhoods of nodes in hypergraphs. We establish their relationships to lattice…
In this paper we develop a structure called Link Algebra, in which we present a Set with two binary operations and an axiom system developed from the study of graph theory and set/antiset theory, sowing main theorems and definitions. Once…
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction we propose algebraic Kan complexes as an…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many…
The theory of $N$-complexes is a generalization of both ordinary chain complexes and graded objects. Hence it yields deeper insight in the structure of these and offers a broader range of applications. This work generalizes the tensor…
Graphs, and sequences of growing graphs, can be used to specify the architecture of mathematical models in many fields including machine learning and computational science. Here we define structured graph "lineages" (ordered by level…
The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.
A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which…
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…
The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose…
We study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and…
We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.
We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez-Dolan slice…
In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…