Cuadrados latinos y códigos secretos: actividades con las que interactuar en el aula de matemáticas

El “fenómeno Sudoku”, iniciado en España en el año 2005, ha alcanzado un éxito destacable. Tratándose de un juego en el que la lógica matemática está en todo momento presente, los Sudokus han logrado una adición que pocos pasatiempos han logrado crear anteriormente. Se han realizado publicaciones expresamente dedicadas únicamente a dicho juego y existen manuales acerca de cómo proceder para resolverlo satisfactoriamente. Sin embargo, no es tan conocido el hecho de que un Sudoku no es más que un caso particular de los denominados cuadrados Latinos, concepto introducido por Euler en 1783 y que en la actualidad ocupa a importantes matemáticos de todo el mundo, debido a las diversas aplicaciones que tienen dichos cuadrados en Economía o Criptografía. Aprovechando este último aspecto, en el presente taller se mostrarán una serie de actividades lúdicas relacionadas con cuadrados Latinos y códigos secretos, las cuales pueden realizarse en el aula de Matemáticas, permitiendo un desarrollo lógico-matemático en el pensamiento del alumnado. Además de la técnica usual en la resolución de Sudokus, estas actividades permiten aunar técnicas de juegos tan conocidos como el “Hundir la flota”, que consiste en descubrir en qué casillas se encuentran los barcos del oponente, o “El Cluedo”, que consiste en descubrir quién es el asesino entre varios posibles candidatos. Por otra parte, el trabajo en equipo es fundamental si se quiere ganar en los juegos que vamos a proponer. En concreto, éste último aspecto fue la razón por la que estas actividades fueron propuestas a los alumnos y alumnas participantes en la fase regional de la XXII Olimpiada Matemática Thales, destinada al alumnado de 2º de E.S.O., organizada por la Sociedad Andaluza de Educación Matemática THALES y que se celebró en Sevilla entre los días 17 y 21 de mayo de 2006

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

Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any isotopism of the former is uniquely related to an isomorphism of the latter. This enables us to develop some results on graph theory in the context of the molecular processes that occur during the S-phase of a mitotic cell cycle. In particular, each monochromatic subset of edges is identified with a mutation or regulatory mechanism that relates any two statuses of the genotypes of a pair of chromatids.Junta de Andalucía FQM-016Junta de Andalucía FQM-32

Algebraic computation of genetic patterns related to three-dimensional evolution algebras

The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic Geometry to determine the distribution of three-dimensional evolution algebras over any field into isotopism classes and hence, to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process. Their distribution into isomorphism classes is also determined in case of dealing with algebras having a onedimensional annihilator

Mutation graphs of asexual diploid organisms

The mutation graph of an asexual diploid organism is introduced as an edgecolouredgraph derived from the genetic pattern by isotopisms of an evolution algebraover a finite field. We describe the step-by-step construction of this graph and establishsome of its basic properties. In order to illustrate this construction, we focus on thespectrum of genetic patterns of two distinct genotypes during a mitosis process

Isotopismos de álgebras de Lie filiformes sobre cuerpos finitos

El presente trabajo trata la distribución del conjunto $\mathcal{F}_n^p$ de álgebras de Lie filiformes de dimensión $n$ sobre $\mathbb{Z}/p\mathbb{Z}$ en clases isomórficas e isotópicas. Se estudian para ello distintas propiedades que deben verificar las aplicaciones lineales correspondientes, al mismo tiempo que se identifica dicho conjunto con la variedad afín asociada a un determinado ideal de polinomios booleanos. Finalmente, para $p=2$, se muestra cómo identificar cada álgebra de $\mathcal{F}_n^2$ con un par de grafos, cuyas clases de isomorfismo pueden identificarse con las clases isomórficas e isotópicas de las correspondientes álgebras

A historical perspective of the theory of isotopisms

In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.Junta de Andalucí

Isotopisms of Lie algebras

The distribution of algebras into equivalence classes is usually done according to the concept of isomorphism. However, such a distribution can also be done into isotopism classes. The concept of isotopism was explicitly introduced in 1942 by Abraham Adrian Albert to classify non-associative algebras. In this poster we deal with the study of isotopisms of Lie algebras. The reasons for using both criteria, isotopisms and isomorphisms, to classify Lie algebras is due to that classifications by isotopisms are different from those by isomorphisms, which involves obtaining new information about these algebras. On a sake of example, we indicate some recent results obtained by ourselves, which are related to the distribution into isomorphism and isotopism classes of filiform Lie algebras over finite fields

Distribución de álgebras de lie, MALCEV y evolución en clases de isotopismos

El presente manuscrito trata distintos aspectos de la teoría de isotopismos de álgebras, centrándose en particular en los isotopismos de álgebras de Lie, de Malcev y de evolución, los cuáles no han sido suficientemente estudiados en la literatura. La distribución que sigue el manuscrito se detalla a continuación. En el Capítulo 1 se expone un breve estudio acerca del origen y desarrollo de la teoría de isotopismos, constituyendo en este sentido la primera introducción en la literatura existente en introducir la mencionada teoría desde un punto de vista general. El Capítulo 2 trata de aquellos resultados en Geometría Algebraica Computacional y en Teoría de Grafos que usamos a lo largo del manuscrito con vistas a determinar computacionalmente las clases de isotopismos de cada tipo de álgebra bajo consideración en los siguientes capítulos. Se describen en particular un par de grafos que permiten definir funtores inyectivos entre álgebras de dimensión finita sobre cuerpos finitos y los citados grafos. El cálculo computacional de invariantes por isomorfismos de estos grafos juega un papel destacable en la distribución de las distintas familias de álgebras en clases de isotopismos y de isomorfismos. Algunos resultados preliminares son expuestos en este sentido, particularmente acerca de la distribución de anillos de cuasigrupos parciales sobre cuerpos finitos. El Capítulo 3 se centra en la distribución de clases de isomorfismos y de isotopismos de dos familias de álgebras de Lie: el conjunto Pn;q de álgebras de Lie prefiliformes n-dimensionales sobre el cuerpo finito Fq y el conjunto Fn(K) de álgebras de Lie filiformes n-dimensionales sobre un cuerpo K. Se prueba concretamente la existencia de n clases de isotopismos en Pn;q. También se introducen dos nuevas series de invariantes por isotopismos que son usados para determinar las clases de isotopismos del conjunto Fn(K) para n≤7 sobre cuerpos algebraicamente cerrados y sobre cuerpos finitos. El Capítulo 4 trata con distintos ideales radicales cero-dimensionales cuyos conjuntos algebraicos asociados pueden indentificarse de forma única con el conjunto Mn(K) de álgebras de Malcev n-dimensionales sobre un cuerpo finito K. El cálculo computacional de sus bases reducidas de Gröbner, junto a la clasificación de álgebras de Lie sobre cuerpos finitos dada por De Graaf y Strade, permiten determinar la distribución de M3(K) y M4(K) no sólo en clases de isomorfismos, que es el criterio usual, sino también en clases de isotopismos. En concreto, probamos la existencia de cuatro clases de isotopismos en M3(K) y ocho clases de isotopismos en M4(K). Además, se prueba que todo álgebra de Malcev 3-dimensional sobre cualquier cuerpo finito y todo álgebra de Malcev 4-dimensional sobre un cuerpo finito de característica distinta de dos es isotópica a un magma-álgebra de Lie. Finalmente, el Capítulo 5 trata con el conjunto En(K) de álgebras de evolución n-dimensionales sobre un cuerpo K, cuya distribución en clases de isotopismos está relacionada de forma única con mutaciones en Genética no Mendeliana. Se centra en concreto en el caso bi-dimensional, el cuál está relacionado con los procesos de reproducción asexual de organismos diploides. Se prueba en particular que el conjunto E2(K) se distribuye en cuatro clases de isotopismos, independientemente de cuál sea el cuerpo base y se caracteriza sus clases de isomorfismos.This manuscript deals with distinct aspects of the theory of isotopisms of algebras. Particularly, we focus on isotopisms of Lie, Malcev and evolution algebras, for which this theory has not been enough studied in the literature. The manuscript is organized as follows. In Chapter 1 we expose a brief survey about the origin and development of the theory of isotopisms. This constitutes a first attempt in the literature to introduce this theory from a general point of view. Chapter 2 deals with those results in Computational Algebraic Geometry and Graph Theory that we use throughout the manuscript in order to compute the isotopism classes of each type of algebra under consideration in the subsequent chapters. We describe in particular a pair of graphs that enable us to define faithful functors between finite-dimensional algebras over finite fields and these graphs. The computation of isomorphism invariants of these graphs plays a remarkable role in the distribution of distinct families of algebras into isotopism and isomorphism classes. Some preliminary results are exposed in this regard, particularly on the distribution of partial-quasigroup rings over finite fields. Chapter 3 focuses on the distribution into isomorphism and isotopism classes of two families of Lie algebras: the set Pn;q of n-dimensional pre- filiform Lie algebras over the finite field Fq and the set Fn(K) of n-dimensional filiform Lie algebras over a base field K. Particularly, we prove the existence of n isotopism classes in Pn;q. We also introduce two new series of isotopism invariants that are used to determine the isotopism classes of the set Fn(K) for n ≤ 7 over algebraically closed fields and finite fields. Chapter 4 deals with distinct zero-dimensional radical ideals whose related algebraic sets are uniquely identified with the set Mn(K) of n-dimensional Malcev magma algebras over a finite field K. The computation of their reduced Gröbner bases, together with the classification of Lie algebras over finite fields given by De Graaf and Strade, enable us to determine the distribution of M3(K) and M4(K) not only into isomorphism classes, which is the usual criterion, but also into isotopism classes. Particularly, we prove the existence of four isotopism classes in M3(K) and eight isotopism classes in M4(K). Besides, we prove that every 3-dimensional Malcev algebra over any finite field and every 4-dimensional Malcev algebra over a finite field of characteristic distinct from two is isotopic to a Lie magma algebra. Finally, Chapter 5 deals with the set En(K) of n-dimensional evolution algebras over a field K, whose distribution into isotopism classes is uniquely related with mutations in non-Mendelian genetics. Particularly, we focus on the two-dimensional case, which is related to the asexual reproduction processes of diploid organisms. We prove that the set E2(K) is distributed into four isotopism classes, whatever the base field is, and we characterize its isomorphism classes

Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field

Evolution algebras were introduced into Genetics to deal with the mechanism of inheritance of asexual organisms. Their distribution into isotopism classes is uniquely related with the mutation of alleles in nonMendelian Genetics. This paper deals with such a distribution by means of Computational Algebraic Geometry. We focus in particular on the twodimensional case, which is related to the asexual reproduction processes of diploid organisms. Specifically, we determine the existence of four isotopism classes, whatever the base field is, and we characterize the corresponding isomorphism classes

Classifications of evolution algebras over finite fields

In this communication we deal with a class of emerging algebras called evolution algebras. These algebras were firstly introduced by J. P. Tian, and then jointly presented with Vojtechovsky in 2006 [6], and later appeared as a book by Tian in 2008 [5]. The motivation to consider these algebras, which lie between algebras and dynamical systems, is due to the fact that at present, the study of them is very extended (see [2, 3, 4], for instance), due to the numerous connections between these algebras and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov Processes, Dynamic Systems and Theory of Knots, among others. An n-dimensional evolution algebra is an algebra (E, •) over a field K which admits a basis {e1, e2, . . . , en} such that ei • ej = 0, if i 6= j, and ei • ei = Pn k=1 aikek, for any 1 ≤ i ≤ n. Regarding these algebras, it is known the classification of evolution algebras into isomorphism classes. At this respect, the main goal of our study is to know the distribution of such algebras in the case of lower dimensions and finite fields, not only into isomorphism classes (the usual criterion), but also into isotopism classes, which constitutes a novel contribution on this subject (a first paper on this topic can be consulted in [1]). To do this we firstly introduce the concept of evolution isotopism as a triple (f, g, h), where f, g and h are non singular linear transformations between E and E 0 such that h(x•y) = f(x)• g(y) for all x, y ∈ E and that for a natural basis {ei}i of E, {f(ei)}i and {g(ei)}i span an evolution subalgebra of E 0 . Next, taking into consideration that the isotopic relation between evolution algebras implies the isomorphic relation, we begin our study starting from the classification of the 2-dimensional complex evolution algebra E. We know that E will be isomorphic to one of the following pairwise non isomorphic algebras E1 : e1 • e1 = e1, E2 : e1 • e1 = e1, e2 • e2 = e1, E3 : e1 • e1 = e1 +e2, e2 • e2 = −e1−e2, E4 : e1•e1 = e2, E5 : e1•e1 = e1+a2e2, e2•e2 = a3e1+e2, 1−a2a3 6= 0, where E5(a2, a3) ' E 0 5 (a3, a2), E6 : e1 • e1 = e2, e2 • e2 = e1 + a4e2, a4 6= 0, where E6(a4) ' E 0 6 (a 0 4 ) ⇔ (a4/a0 4 ) = cos (2πk/3) + i sin (2πk/3), for some k = 0, 1, 2. This classification allows us to study the isotopic relations between these classes. Indeed, we find that algebra E1 is isotopic to algebra E4, E2 to E3, E5 to E6, and we also obtain that algebras E1, E2 and E5 are not isotopic. Finally, we complete this work with the study of the classifications of low dimensional evolution algebras over Z/pZ, with p prime. We find that the number of isomorphic (or isotopic) classes depends on the dimension of the algebra and on the the value of the integer p