Quasigroups constructed from perfect Mendelsohn designs with block size 4

摘要

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 designs PMD(9, 4) and PMD(12, 4) as well as for (nonperfect) Mendelsohn designs MD(8, 4) are given.