A unified theory for coherent systems in reliability II. A unified viewpoint on multinary coherent systems through monotone coherence and relevance

Binary coherent system theory has played an important part in reliability. Its extension to (‘degradable’ or ‘multistate’ or) multinary systems has recently been considered in various papers, through various definitions. This paper studies the most general model for multinary systems, proposes a unified viewpoint on multinary coherent systems and gives unified arguments to apply and to investigate further the binary and multinary cases. In a more detailed way, the ‘helpful bridge’ lately proposed by Block and Savits1 between the binary and multinary cases is completed and multinary systems then can be studied in terms of monotone binary coherent systems, introduced in a companion paper.2 Through various results, multinary systems are examined in terms of structure functions and of life functions; fundamental relations for their analysis are obtained with their set characterizations; the main axis that can be retained among the numerous types of coherence is emphasized, in a unified way, through relevance; reliability models are examined through performance processes, life lengths and performance functions; and Birnbaum's factors of importance are thoroughly extended to the multinary case. Fundamental results proposed in previous studies are thus completed with a shorter unified approach.