Orthogonality and retract orthogonality of operations

Abstract

In this article, we study the connections between orthogonality and retract orthogonality of operations. We prove that if a tuple of operations is retractly orthogonal, then it is orthogonal. However, the orthogonality of operations doesn’t provide their retract orthogonality. Consequently, every k-tuple of orthogonal k-ary operations is prolongable to a k-tuple of orthogonal n-ary operations. Also, we give some specifications for central quasigroups. In particular for central quasigroups over finite field of prime order, retract orthogonality is the necessary and sufficient condition for orthogonality. The problem of coincidence of orthogonality and retract orthogonality remains open.