On quasigroups satisfying Stein's third law

Griggs, Terry S. and Kozlik, Andrew R. (2021). On quasigroups satisfying Stein's third law. Discrete Mathematics, 344(10), article no. 112526.

DOI: https://doi.org/10.1016/j.disc.2021.112526


A quasigroup (Q, ·) of order v satisfies Stein’s third law if (y ·x) ·(x ·y) =x holds for all x, y ∈Q. Let the quasigroup contain n idempotent elements. We construct such quasigroups with (v, n) ∈\{(20, 0), (24, 0), (28, 0), (36, 0)\}, thus completing the existence spectrum of quasigroups satisfying Stein’s third law with no idempotents. We also construct previously unknown quasigroups with (v, n) ∈\{(17, 11), (21, 3), (21, 7), (24, 4), (25, 7), (25, 19)\} and provide an enumeration for all v ≤9.

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