On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations

In this paper we study systems of fuzzy relation inequalities and equations of the formU ◦Vi 6 Vi ◦U (i ∈ I), where U is an unknown and Vi (i ∈ I) are given fuzzy relations, the dual systems Vi ◦ U 6 U ◦ Vi (i ∈ I), their conjunctions, the systems of the formU◦Vi = Vi◦U (i ∈ I), and certain special types of these systems. We call themweakly linear systems. For each weakly linear system, with a complete residuated lattice as the underlying structure of truth values, we prove the existence of the greatest solution, and we provide an algorithm for computing the greatest solution, which work whenever the underlying complete residuated lattice is locally finite. Otherwise, we determine some sufficient conditions under which the algorithm works. The algorithm is iterative, and each its single step can be viewed as solving a particular linear system. Weakly linear systems emerged from the fuzzy automata theory, but we show that they also have important applications in other fields, e.g. in the concurrency theory and social network analysis.