Quasigroups constructed from perfect Mendelsohn designs with block size 4

Griggs, Terry S.; Drápal, Aleš and Kozlik, Andrew R. (2020). Quasigroups constructed from perfect Mendelsohn designs with block size 4. Journal of Combinatorial Designs, 28(7) pp. 489–508.

DOI: https://doi.org/10.1002/jcd.21708

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

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