Cycle structures of autotopisms of the Latin squares of order up to 11

The cycle structure of a Latin square autotopism Θ = (α, β, γ) is the triple (lα, lβ, lγ), where lδ is the cycle structure of δ, for all δ ∈ {α, β, γ}. In this paper we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to 11

The set of autotopisms of partial Latin squares

Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $\Theta$, the cardinality of the set $\mathcal{PLS}_{\Theta}$ of partial Latin squares which are invariant under $\Theta$ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties studied. It is also seen that the cycle structure of $\Theta$ determines the possible sizes of the elements of $\mathcal{PLS}_{\Theta}$ and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square.Comment: 20 pages, 4 table

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

Determinants of latin squares of a given pattern

Cycle structures of autotopisms of Latin squares determine all possible patterns of this kind of design. Moreover, given any isotopism, the number of Latin squares containing it in their autotopism group only depends on the cycle structure of this isotopism. This number has been studied in for Latin squares of order up to 7, by following the classification given in. Specifically, regarding each symbol of a Latin square as a variable, any Latin square can be seen as the vector space associated with the solution of an algebraic system of polynomial equations, which can be solved using Gröbner bases, by following the ideas implemented by Bayer to solve the problem of n-colouring a graph. However, computations for orders higher than 7 have been shown to be very difficult without using some other combinatorial tools. In this sense, we will see in this paper the possibility of studying the determinants of those Latin squares related to a given cycle structure. Specifically, since the determinant of a Latin square can be seen as a polynomial of degree n in n variables, it will determine a new polynomial equation that can be included into the previous system. Moreover, since determinants of Latin squares of order up to 7 determine their isotopic classes, we will study the set of isotopic classes of Latin squares of these orders related to each cycle structure