Publication at Faculty of Mathematics and Physics |
2020
Abstract
Several varieties of quasigroups obtained from perfect Mendelsohn designs with block size 4 are defined.One of these is obtained from the so-called directed standard construction and satisfies the law xy*(y*xy)=x and another satisfies Stein's third law xy*yx=y. Such quasigroups which satisfy the flexible law x*yx=xy*x are investigated and characterized.
Quasigroups which satisfy both of the laws xy * (y *xy) = x and xy * yx = y are shown to exist.Enumeration results for perfect Mendelsohn design PMD(9, 4) and PMD(12, 4) as well as for (nonperfect) Mendelsohn designs MD(8, 4) are given.