Half quasigroups and generalized quasigroup orthogonality

It is still unknown whether three mutually orthogonal Latin squares (resp. quasigroups) of order 10 exist or whether there is a check digit system of order 10 which detects all twin errors. During our research on these topics we use an approach with half quasigroups, which leads to an interesting generalization of quasigroup orthogonality. A (vertical) half quasigroup ( H , ∗ ) is a groupoid for which the right cancellation law x ∗ y = x ′ ∗ y ⇒ x = x ′ holds. It is close related to what is known as row or column Latin square. The set of all half quasigroups H n of order n together with an operation ⋅ builds a group ( H n , ⋅ ) and the set of quasigroups Q n is a subset of H n . Two half quasigroups h , g ∈ H n are orthogonal if and only if a quasigroup q ∈ Q n exists with h ⋅ q = g . We show that this is just a special case and can be generalized to arbitrary groups. rmore, we prove a conjecture of Dénes, Mullen and Suchower about Latin power sets by showing that for all orders n ≠ 2 , 6 there is a quasigroup q of order n with q 2 ∈ Q n and q is orthogonal to q 2 . Moreover, a computer search verifies a result of Wanless that there is no quasigroup q of order 10 having q 2 and q 3 ∈ Q 10 .