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,∗)(H,∗) is a groupoid for which the right cancellation law x∗y=x′∗y⇒x=x′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 HnHn of order nn together with an operation ⋅⋅ builds a group (Hn,⋅)(Hn,⋅) and the set of quasigroups QnQn is a subset of HnHn. Two half quasigroups h,g∈Hnh,g∈Hn are orthogonal if and only if a quasigroup q∈Qnq∈Qn exists with h⋅q=gh⋅q=g. We show that this is just a special case and can be generalized to arbitrary groups.Furthermore, we prove a conjecture of Dénes, Mullen and Suchower about Latin power sets by showing that for all orders n≠2,6n≠2,6 there is a quasigroup qq of order nn with q2∈Qnq2∈Qn and qq is orthogonal to q2q2. Moreover, a computer search verifies a result of Wanless that there is no quasigroup qq of order 1010 having q2q2 and q3∈Q10q3∈Q10.