On Quasigroups Satisfying Stein'S Third Law

A quasigroup (Q, .) of order v satisfies Stein's third law if (y . x) . (x . y) = x holds for all x, y is an element of Q. Let the quasigroup contain n idempotent elements. We construct such quasigroups with (v, n) is an element of {(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) is an element of {(17, 11), (21, 3), (21, 7), (24, 4), (25, 7), (25, 19)} and provide an enumeration for all v <= 9. (C) 2021 Elsevier B.V. All rights reserved.