Meet-continuous codomain lattice and fuzzy set inequations

Abstract

Extremal problems in fuzzy set and fuzzy relational equations and inequations - both existential and constructional - strongly depend on the choice of the codomain lattice, and the way we define the composition of fuzzy relations. A complete lattice is meet-continuous if infimum commutes with the supremum of chains. For such a lattice, I prove that some inequations and equations that are often studied have a maximal solution. If the existing maximal solution is unique, it is also the greatest solution. Solutions to some inequations are connected to the notions of fuzzy relations closed under composition with a given fuzzy relations, thus we prove the existence of maximal such relations. The property of being transitive relation may also be expressed in terms of being a solution to a fuzzy relational inequation, thus I also prove the existence of a maximal transitive relation contained in a given arbitrary relation. In case of complete codomain lattice, there exists the least solution to some fuzzy relational inequations. If such complete codomain lattice is also meet-continuous, there exist algorithms giving such least solutions in at most countably many sets. In particular, there exists an algorithm for finding the transitive closure of an arbitrary fuzzy relation in at most countably many steps. Even if the number of steps is infinite, in some cases, by inductive reasoning, we shall be able to determine the transitive closure in real time, as an example illustrates.