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

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

Presemifields, bundles and polynomials over GF (pn)

The content of this thesis is first and foremost about presemifields and the equivalence classes they may be categorized by. This equivalence has been termed “bundle equivalence'' by Horadam. Bundle equivalence is inherited from multiplicative orthogonal cocycles, and the final Chapter is devoted entirely to coboundaries and cocycles. In this thesis we provide a complete computational classification of the bundles of presemifields in all presemifield isotopism classes of order p n , provide a formula for the number of bundles in the presemifields isotopism class of GF (p 2 ) and give a representative of each bundle, for any prime p . We provide computational classification of the bundles of presemifields in the isotopism class of GF (p 3 )  for the cases  p =3,5,7,11 and give representatives, give formulae for two of the three possible size bundles in the presemifield isotopism class of  GF (p 3 )   which we call the minimum and the mid-size bundles. We provide a Conjecture which states the total number of mid-size bundles in the isotopism class of  GF (p 3 ) and give a computational classification of the bundles of presemifields in the isotopism class of  GF (2 5 ) and  GF (3 4 ) . We provide a measurement of the differential uniformity of functions derived from the diagonal map of presemifield multiplications with order p n < 16 and derive bivariate polynomial formulae for cocycles and coboundaries in We produce a basis for the ( p n - 1 - n ) - dimensional -space of coboundaries. When p = 2 we give a recursive definition of the basis coboundaries. We use the Kronecker product to explain the self-similarity of the binomial coefficients modulo a prime and use the Kronecker product to define recursively the basis coboundaries for p odd, and we demonstrate this holds for the case p = 2. We show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form when  p = 2.  The results of this thesis have been published in the Proceedings of the International Workshop on Coding and Cryptography, Designs, Codes and Cryptography and the Proceedings of IEEE International Symposium on Information Theory and will appear in the Journal of the Australian Mathematical Society

On Loop Commutators, Quaternionic Automorphic Loops, and Related Topics

This dissertation deals with three topics inside loop and quasigroup theory. First, as a continuation of the project started by David Stanovský and Petr Vojtĕchovský, we study the commutator of congruences defined by Freese and McKenzie in order to create a more pleasing, equivalent definition of the commutator inside of loops. Moreover, we show that the commutator can be characterized by the generators of the inner mapping group of the loop. We then translate these results to characterize the commutator of two normal subloops of any loop. Second, we study automorphic loops with the desire to find more examples of small orders. Here we construct a family of automorphic loops, called quaternionic automorphic loops, which have order 2n for n ≥ 3, and prove several theorems about their structure. Although quaternionic automorphic loops are nonassociative, many of their properties are reminiscent of the generalized quaternion groups. Lastly, we find varieties of quasigroups which are isotopic to commutative Moufang loops and prove their full characterization. Moreover, we define a new variety of quasigroups motivated by the semimedial quasigroups and show that they have an affine representation over commutative Moufang loops similar to the semimedial case proven by Kepka

Integer and Constraint programming methods for mutually Orthogonal Latin Squares.

This thesis examines the Orthogonal Latin Squares (OLS) problem from the viewpoint of Integer and Constraint programming. An Integer Programming (IP) model is proposed and the associated polytope is analysed. We identify several families of strong valid inequalities, namely inequalities arising from cliques, odd holes, antiwebs and wheels of the associated intersection graph. The dimension of the OLS polytope is established and it is proved that certain valid inequalities are facet-inducing. This analysis reveals also a new family of facet-defining inequalities for the polytope associated with the Latin square problem. Separation algorithms of the lowest complexity are presented for particular families of valid inequalities. We illustrate a method for reducing problem's symmetry, which extends previously known results. This allows us to devise an alternative proof for the non-existence of an OLS structure for n = 6, based solely on Linear Programming. Moreover, we present a more general Branch & Cut algorithm for the OLS problem. The algorithm exploits problem structure via integer preprocessing and a specialised branching mechanism. It also incorporates families of strong valid inequalities. Computational analysis is conducted in order to illustrate the significant improvements over simple Branch & Bound. Next, the Constraint Programming (CP) paradigm is examined. Important aspects of designing an efficient CP solver, such as branching strategies and constraint propagation procedures, are evaluated by comprehensive, problem-specific, experiments. The CP algorithms lead to computationally favourable results. In particular, the infeasible case of n = 6, which requires enumerating the entire solution space, is solved in a few seconds. A broader aim of our research is to successfully integrate IP and CP. Hence, we present ideas concerning the unification of IP and CP methods in the form of hybrid algorithms. Two such algorithms are presented and their behaviour is analysed via experimentation. The main finding is that hybrid algorithms are clearly more efficient, as problem size grows, and exhibit a more robust performance than traditional IP and CP algorithms. A hybrid algorithm is also designed for the problem of finding triples of Mutually Orthogonal Latin Squares (MOLS). Given that the OLS problem is a special form of an assignment problem, the last part of the thesis considers multidimensional assignment problems. It introduces a model encompassing all assignment structures, including the case of MOLS. A necessary condition for the existence of an assignment structure is revealed. Relations among assignment problems are also examined, leading to a proposed hierarchy. Further, the polyhedral analysis presented unifies and generalises previous results