Transitive reduction of a nilpotent boolean matrix

Boolean matrices are widely used in many fields, and the theory of boolean matrices is related to the algebra of relations, switching theory, and graph theory. First some basic properties of nilpotent matrices are shown in the paper. A nilpotent boolean matrix plays an important role in the theory of boolean matrices. The purpose is to present those properties of boolean matrices which are related to finding the transitive reduction of a nilpotent matrix, or an acyclic graph.