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

Interconnecting two multilevel systems which are defined on the same transitive contextual relation is a problem encountered in structuring complex systems using the interpretive structural modelling process. 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.