Fuzzy relation equations and subsystems of fuzzy transition systems

In this paper we study subsystems, reverse subsystems and double subsystems of a fuzzy transition system. We characterize them in terms of fuzzy relation inequalities and equations, as eigen fuzzy sets of the fuzzy quasi-order Qδ and the fuzzy equivalence Eδ generated by fuzzy transition relations, and as linear combinations of aftersets and foresets of Qδ and equivalence classes of Eδ. We also show that subsystems, reverse subsystems and double subsystems of a fuzzy transition systemT form both closure and opening systems in the lattice of fuzzy subsets of A, where A is the set of of states of T , and we provide efficient procedures for computing related closures and openings of an arbitrary fuzzy subset of A. These procedures boil down to computing the fuzzy quasi-order Qδ or the fuzzy equivalence Eδ, which can be efficiently computed using the well-known algorithms for computing the transitive closure of a fuzzy relation.