Two types of ordinal sums of fuzzy implications on bounded lattices

Abstract

In this paper, two kinds of ordinal sums of fuzzy implications on bounded lattices are provided. The first one is complementing a specific fuzzy implication to a given family of fuzzy implications defined on pairwise disjoint closed subintervals of a bounded lattice. The second way is defining specific values outside the given family of countably many subintervals, where the endpoints of the subintervals constitute a chain. For the first way, necessary and sufficient conditions for a fuzzy implication to be a complement of a given family of fuzzy implications such that the ordinal sum is a fuzzy implication are provided, which generalize some results of fuzzy implications on the real unit interval. Based on this, we deal with those elements that are incomparable with the endpoints of the given subintervals and provide a fuzzy implication as a complement such that for arbitrary family of fuzzy implications on the given subintervals, the ordinal sum is always a fuzzy implication. For the second ordinal sum, we provide two methods for defining values outside the given subintervals that are shown to be always fuzzy implications.