Two versions of Tarski fixed point theorem applied to fuzzy set equations and inequations

Abstract

The set of all fuzzy subsets of a given set and the set of all fuzzy relations on a given set are complete lattices, provided that the codomain lattice is complete. That gives us an opportunity to apply Tarski fixed point theorem to some typical fuzzy set and fuzzy relational equations, the solutions to which are the fixed points of a monotonous operator. Moreover, the solutions to some systems of fuzzy set equations may also be seen as the fixed points of a monotonous operator on a complete lattice. Using the original version of Tarski fixed point theorem we solve some existential and extremal problems. Using its constructive version we get even more, a “construction” of the existing solution. Such a construction may take uncountably many steps, thus it is proved that in a special case of a meet-continuous lattice, the process may end in at most countably many steps.