On quasigroups satisfying Stein's third law

Abstract 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 .