A theorem and a procedure for the complete implication matrix of system interconnection matrices

Interconnecting two multilevel systems which are defined on the same contextual relation that is transitive is a problem encountered in structuring complex systems using the interpretive structural modeling (ISM) process. The theory of interconnection is an integral part of ISM. The present report is concerned with the results of subsequent detailed research. A theorem that states that the complete implication matrix can be obtained by squaring the adjacency implication matrix is proved. Then a procedure for calculating the complete implication matrix in O((mn)
²
) time by means of the results obtained through the theorem is presented, where m and n are the number of element of each system. A simple expression of the complete implication matrix for transitive bordering is also given.