Invertibility, Equivalence, and the Decomposition Property of Abstract Systems

The concepts of inverse and left inverse systems, weakly equivalent and strongly equivalent systems, and discretetime and finite memory systems are examined. Various relations between these concepts are found, and the role of the decomposition property of the system input-output state relation is investigated. Among the results obtained are 1) under suitable controllability assumptions, left inverse systems are inverse systems, 2) weakly equivalent finite memory systems with the decomposition property are strongly equivalent, and 3) finite-state systems with the decomposition property have finite memory.