Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver

Conjugacy and orthogonality of Latin squares have been widely studied in the literature not only for their theoretical interest in combinatorics, but also for their applications in distinct fields as experimental design, cryptography or code theory, amongst others. This paper deals with a series of binary constraints that characterize the sets of partial Latin squares of a given order for which their six conjugates either coincide or are all of them distinct and pairwise orthogonal. These constraints enable us to make use of a SAT solver to enumerate both sets. As an illustrative application, it is also exposed a method to construct totally symmetric partial Latin squares that gives rise, under certain conditions, to new families of Lie partial quasigroup rings

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$

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

Embedding partial totally symmetric quasigroups

This paper concerns the embedding problem for partial totally symmetric quasigroups. For all it >= 9, it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and u >= 2n + 4, and this is the best possible such inequality. (c) 2006 Elsevier Inc. All rights reserved