相关论文: Algebra generators and information theory

We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of…

To constitute atoms of a $\sigma$ algebra is not a easy task due to the large number of its elements. However, determining them via generators seems a feasible and simple way since most $\sigma$ algebras are generated by their smaller…

We develop an analogue of universal algebra in which generating symbols are interpreted as relations. We prove a variety theorem for these relational algebraic theories, in which we find that their categories of models are precisely the…

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to the one of equational classes defined by equation arrows. Free…

This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the…

In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all 2-generated…

We begin by defining Temperley-Lieb algebra, in two different ways: as a presented algebra or as a diagrammatic algebra. Next, we look for a basis algorithmically, using rewriting theory. Finally, we introduce a generalization of the…

An algebraic formalism for the study of interacting particle systems is developed. Particle processes are described in terms of the category theory. The problem for the unique description of these processes is discussed. Categories relevant…

The problem of quantizing theories defined over configuration spaces described by non-commuting parameters is considered. In this paper we describe the first step in this direction, that is the definition of an integral over a general…

A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory…

The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…

Due to the increased complexity of software development projects more and more systems are described by models. The sheer size makes it impractical to describe these systems by a single model. Instead many models are developed that provide…

Abstract clones serve as an algebraic presentation of the syntax of a simple type theory. From the perspective of universal algebra, they define algebraic theories like those of groups, monoids and rings. This link allows one to study the…

We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

The objective of this paper is to determine the finite dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to…

The primary goal of this paper is to present a unified way to transform the syntax of a logic system into certain initial algebraic structure so that it can be studied algebraically. The algebraic structures which one may choose for this…

We first provide an overview of several results dealing with the genus of a division algebra and highlight the role of ramification in its analysis. We then give a survey of recent developments on the genus problem for simple algebraic…

This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…