10.18453/rosdok_id00005832 Gronau, Hans-Dietrich Hans-Dietrich Gronau http://d-nb.info/gnd/138612498 Schmidmeier, Markus Markus Schmidmeier http://d-nb.info/gnd/173040292 https://orcid.org/0000-0003-3365-6666 University of Rostock, Institute of Mathematics 2007 510 Mathematics 2007 en https://purl.uni-rostock.de/rosdok/id00005832 urn:nbn:de:gbv:28-rosdok_id00005832-5 Let K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian.