On a class of commutative groupoids determined by their associativity triples
Let be a commutative groupoid such that ; ; or . Then is determined uniquely up to isomorphism and if it is finite, then for an integer .
On a distance of groups and Latin squares
On algebras of generalized Latin squares
The main result of this paper is the introduction of a notion of a generalized -Latin square, which includes as a special case the standard Latin square, as well as the magic square, and also the double stochastic matrix. Further, the algebra of all generalized Latin squares over a commutative ring with identity is investigated. Moreover, some remarkable examples are added.
On an algorithm for a double-resolvability test.
On approximation of amenable groups by finite quasigroups.
On Balanced Room Squares and Complete Balanced Howell Rotations
On biembeddings of Latin squares.
On certain regular graphs of girth 5.
On Difference Matrices, Resolvable Transversal Designs and Generalized Hadamard Matrices.
On elementary moves that generate all spherical latin trades
We show how to generate all spherical latin trades by elementary moves from a base set. If the base set consists only of a single trade of size four and the moves are applied only to one of the mates, then three elementary moves are needed. If the base set consists of all bicyclic trades (indecomposable latin trades with only two rows) and the moves are applied to both mates, then one move suffices. Many statements of the paper pertain to all latin trades, not only to spherical ones.
On mixed codes with covering radius 1 and minimum distance 2.
On multiplicative magic squares.
On orthogonal Latin -dimensional cubes
We give a construction of orthogonal Latin -dimensional cubes (or Latin hypercubes) of order for every natural number and . Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker.
On small distances of small 2-groups
The paper reports the results of a search for pairs of groups of order that can be placed in the distance for the case when . The constructions that are used are of the general character and some of their properties are discussed as well.
On some Generalizations of a Class of Discrete Functions
In this paper we examine discrete functions that depend on their variables in a particular way, namely the H-functions. The results obtained in this work make the “construction” of these functions possible. H-functions are generalized, as well as their matrix representation by Latin hypercubes.
On the completion of incomplete latin squares.
On the existence of -SOLSSOMs.
On the existence of MOLS with equal-sized holes.
On the Existence of Room Squares
On the Existence of Room Squares of Order 4n