Computation of isotopisms of algebras over finite fields by means of graph invariants

Abstract

In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine whether two algebras are isomorphic. In order to illustrate their efficiency, we determine explicitly the classification of two-and threedimensional partial quasigroup rings.

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

“…As an illustrative application of the exposed study, we also delve into a recent work developed by the authors 37 about the enumeration of partial quasigroup rings over finite fields derived from partial Latin squares. Bruck 15 introduced the concept of quasigroup ring related to a quasigroup (S, ·) as an algebra of basis {e a |a ∈ S} over a base field K such that e a e b = e a·b , for all a, b ∈ S. This concept is straightforwardly generalized to that of partial quasigroup ring in case of being the pair (S, ·) a partial quasigroup.…”

Computational analysis of LED circuits based on partial quasi‐group rings

“…If there are not empty cells, then this constitutes a Latin square of order n. In such a case, the pair (Q, ·) is, in turn, a quasi-group. [43][44][45] The distribution of Latin squares into isomorphism classes is known for order n ≤ 11, [37][38][39] , and that of partial Latin squares has been explicitly computed for order n ≤ 6.…”

“…Different types of partial quasi-group rings have been studied from a computational point of view. [43][44][45]…”

An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics

“…Specifically, we introduce a total-colored graph that can be associated with any given evolution algebra over a finite field so that any isomorphism of the former is uniquely related to an isotopism of the latter. The underlying idea behind the proposed graph derives from a previous work of the authors [29] in which a pair of colored graphs was introduced in order to describe faithful functors relating the category of finite-dimensional algebras over finite fields with the category of vertex-colored graphs. They were based in turn on a proposal of McKay et al [30], who identified the isotopisms of Latin squares with isomorphisms of vertex-colored graphs.…”

“…They were based in turn on a proposal of McKay et al [30], who identified the isotopisms of Latin squares with isomorphisms of vertex-colored graphs. A first attempt to approach the graphs introduced in [29] to the theory of evolution algebras was carried out in [31], where a step-by-step construction of an edge-colored graph derived from the genetic pattern of an evolution algebra over a finite field was established. The total-colored graph here introduced not only simplifies this last construction by reducing both the number of vertices and edges under consideration, but also facilitates a relationship among different concepts on evolution algebras, graph theory, and genetics.…”