A digraph is Vosperian if any fragment has cardinality one or \(| V(X)|-d^ +(X)-1\). A digraph is superconnected if every minimum cutset is the set of vertices adjacent from or to some vertex. The authors use a result of \textit{J. H. B. Kemperman} [Acta Math. 103, 63--88 (1960; Zbl 0108.25704)] to characterize Vosperian and superconnected abelian Cayley directed graphs, including loop networks (also called circulants).