Modified Ordinal Sums of Triangular Norms and Triangular Conorms on Bounded Lattices

Abstract

Having in mind that ordinal sum construction of triangular norms (triangular conorms) may not work on bounded lattices, in general, we propose a modification of ordinal sums of t‐norms (t‐conorms) resulting to a t‐norm (t‐conorm) on an arbitrary bounded lattice. In particular, our method can be applied to define connectives for fuzzy sets type 2, interval‐valued fuzzy sets, intuitionistic fuzzy sets, etc. Some illustrative examples are added, considering the interval lattice L([0, 1]), the intuitionistic lattice LI, and the diamond lattice.

Ordinal sums of triangular norms on a bounded lattice

“…The key lies in whether it always leads to a t-norm. In this paper, appropriately dealing with those elements that are incomparable with the endpoints of the given subintervals, by synthesizing the techniques of [3] and [9], we propose an alternative definition of ordinal sum of countably many t-norms on subintervals of a complete lattice, where the endpoints of the subintervals constitute a chain.…”

A way to obtain 2-uninorm on bounded lattice from uninorms defined on subintervals of bounded lattice

“…Since bounded lattice case is more complex, uninorms on bounded lattices has been a challenging problem for many researchers [3,5,6,7,11]. Besides uninorm on [0, 1], 2-uninorms are defined and studied [1,2,4].…”

Ordering based 2-uninorm on bounded lattice

“…Because of applications of uninorm like fuzzy logic, expert systems, neural networks, fuzzy system modelling [6,16], it is attracted interest. The generalization of uninorm to a complete lattice has been an challenging problem for many researchers [2,4,10,14]. The order from logical operation has gained interest [11,12,13] in recent years.…”