Isaías David Marín Gaviria

We study g-digroup actions by proving some results that are well known in group actions, for example: the orbit stabilizer theorem, the class equation, Burnside’s formula, and we relate g-digroup actions with trioids. We introduce the definition of a representation of a g-digroup into dialgebras and establish a strong connection with g-digroup action, like in the group case.

Representation group theory plays an important role in group theory, much more in geometry, on which we can determine, in many cases, what kind of geometry or geometries a manifold M supports by knowing a faithful representation of the fundamental group Π1(M) in the group Iso + (M) of preserving-orientations isometries of M. Motivated by this and by the definition of a g-digroup action given in [1], in this paper we present a study of such action by proving some interesting results that are well known in group actions, for example: the orbit stabilizer theorem, the class equation, Burnside’s formula and we relate g-digroup actions with trioids. We introduce the definition of representation of a g-digroup into dialge-bras and a strong connection with g-digroup action, like in the case of group. We present the g-digroups as an invariant of quotient orbifolds.

Currently, researching the Yang–Baxter equation (YBE) is a subject of great interest among scientists of diverse areas in mathematics and other sciences. One of the fundamental open problems is to find all of its solutions. The investigation deals with developing theories such as knot theory, Hopf algebras, quandles, Lie and Jordan (super) algebras, and quantum computing. One of the most successful techniques to obtain solutions of the YBE was given by Rump, who introduced an algebraic structure called the brace, which allows giving non-degenerate involutive set-theoretical solutions. This paper introduces Brauer configuration algebras, which, after appropriate specializations, give rise to braces associated with Thompson’s group F. The dimensions of these algebras and their centers are also given.

The enumeration of Dyck paths is one of the most remarkable problems in Catalan combinatorics. Recently introduced categories of Dyck paths have allowed interactions between the theory of representation of algebras and cluster algebras theory. As another application of Dyck paths theory, we present Brauer configurations, whose polygons are defined by these types of paths. It is also proved that dimensions of the induced Brauer configuration algebras and the corresponding centers are given via some integer sequences related to Catalan triangle entries.

We introduce an algorithm based on posets and tiled orders to generate emerging images. Experimental results allow concluding that images obtained with these kinds of tools are easy to detect by human beings. It is worth pointing out that the emergence phenomenon is a Gestalt grouping law associated with AI open problems. For this reason, emerging images have arisen in the last few years as a tool in the context of the development of human interactive proofs.

Mutations on Brauer configurations are introduced and associated with some suitable automata to solve generalizations of the Chicken McNugget problem. Additionally, based on marked order polytopes, the new Diophantine equations called Gelfand–Tsetlin equations are also solved. The approach allows algebraic descriptions of some properties of the AES key schedule via some Brauer configuration algebras and suitable non-deterministic finite automata (NFA).