Sheila Oates-Williams

On the variety generated by Murskii's algebra

It is shown that the variety generated by Murskiis algebra contains uncountably many subvarieties.

On the distance between distinct group Latin squares

In an article in 1992, Drapal addressed the question of how far apart the multiplication tables of two groups can be? In this article we continue this investigation; in particular, we study the interaction between partial equalities in the multiplication tables of the two groups and their subgroup structure

Single laws for sloops and squags

Abstract We give single laws for the variety of all sloops and the variety of all squags.

Balanced binary arrays iii: the hexagonal grid

We consider the following problem arising in agricultural statistics. Suppose that a large number of plants are set out on a regular grid, which may be triangular, square or hexagonal, and that among these plants, half are to be given one and half the other of two possible treatments. For the sake of statistical balance, we require also that, if one plant in every k plants has i of its immediate neighbours receiving the same treatment as itself, then k is constant over all possible values of i. For square and triangular grids, there exist balanced arrays of finite period in each direction, but for the hexagonal grid, we show that no such balanced array can exist. Several related questions are discussed.

Strongly 2-perfect cycle systems and their quasigroups

A recent result of Bryant and Lindner shows that the quasigroups arising from 2-perfect m-cycle systems form a variety only when m = 3, 5 and 7. Here we investigate the situation in the case where the distance two cycles are required to be in the original system.

Constructing identities for finite quasigroups

An algorithm for producing identities which hold in any given finite quasigroup is described. Identities produced by the algorithm are used to prove several results concerning varieties of quasigroups. In particular varieties of quasigroups associated with various combinatorial designs are examined.

Mendelsohn designs associated with a class of idempotent quasigroups

Abstract The groupoid operation defined by x ∗ y= -μx + (1 + μ)y on finite fields was used by Mendelsohn to construct cyclic designs. We investigate the more general situation where the underlying structure is Z n , with n odd, but not necessarily prime.