603 research outputs found
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
Some new conjugate orthogonal Latin squares
AbstractWe present some new conjugate orthogonal Latin squares which are obtained from a direct method of construction of the starter-adder type. Combining these new constructions with earlier results of K. T. Phelps and the first author, it is shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal Latin square of order v exists for all positive integers v ≠2, 6. It is also shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal idempotent Latin square of order v exists for all positive integers v ≠2, 3, 6 with one possible exception v = 12, and this result can be used to enlarge the spectrum of a certain class of Mendelsohn designs and provide better results for problems on embedding
Nonexistence of Certain Skew-symmetric Amorphous Association Schemes
An association scheme is amorphous if it has as many fusion schemes as
possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V.
Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of
combinatorial objects, pages 39--49. VNIISI, Moscow, Institute for System
Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A.
Munemasa and M. Yamada [T. Ito, A. Munemasa and M. Yamada, Amorphous
association schemes over the Galois rings of characteristic 4, European J.
Combin., 12(1991), 513--526]. A scheme is called skew-symmetric if the diagonal
relation is the only symmetric relation. We prove the nonexistence of
skew-symmetric amorphous schemes with at least 4 classes. We also prove that
non-symmetric amorphous schemes are commutative.Comment: 10 page
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 AndalucÃ
2011 - The Sixteenth Annual Symposium of Student Scholars
The full program book from the Sixteenth Annual Symposium of Student Scholars, held on April 26, 2011. Includes abstracts from the presentations and posters.https://digitalcommons.kennesaw.edu/sssprograms/1010/thumbnail.jp
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