There are asymptotically the same number of Latin squares of each parity

A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order n there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order n → ∞

Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$

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

Parity of Sets of Mutually Orthogonal Latin Squares

Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an $\mathrm{OA}(k,n)$ has an information content of $\dim(k,n)$ bits. We show that $\dim(k,n) \leq {k \choose 2}-1$. For the case corresponding to projective planes we prove a tighter bound, namely $\dim(n+1,n) \leq {n \choose 2}$ when $n$ is odd and $\dim(n+1,n) \leq {n \choose 2}-1$ when $n$ is even. Using the existence of MOLS with subMOLS, we prove that if $\dim(k,n)={k \choose 2}-1$ then $\dim(k,N) = {k \choose 2}-1$ for all sufficiently large $N$. Let the ensemble of an $\mathrm{OA}$ be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an $\mathrm{OA}(k,n)$ can contain. These restrictions depend on $n\mod4$ and give some insight as to why it is harder to build projective planes of order $n \not= 2\mod4$ than for $n \not= 2\mod4$. For example, we prove that when $n \not= 2\mod 4$ it is impossible to build an $\mathrm{OA}(n+1,n)$ for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols)

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