On the graphs of finite idempotent Boolean relation matrices

This paper presents a graph-theoretic characterization of idempotent Boolean relation matrices of finite order. A relation-theoretic point of view is adopted in the paper. Idempotent matrices appear in the sequence of powers of any Boolean relation matrix, and are of purely theoretical as well as applied interest in connection with issues of convergence. The results provide a detailed description of the connectivity and cyclic structure of the directed graphs of idempotent matrices. The study is basically motivated by certain connectivity and flow problems which arise in the analysis of largescale information systems. The formal results are exemplified in an investigation of the asymptotic forms of a recursive model of an information system which affords a conjoin t representation of processes of communication and derivation of information. A second principal application is given in a process formulation for the generation of cons istent rank orderings. The relation between system des ign and idempotent forms is exhibited in the two applications.