Lattice Generalizations of the Concept of Fuzzy Numbers and Zadeh's Extension Principle

TL;DR

This paper generalizes fuzzy numbers to lattice-structured sets, modifies Zadeh's extension principle accordingly, and explores applications in cognitive maps and expert assessment comparisons.

Contribution

It introduces a lattice-based generalization of fuzzy numbers and adapts Zadeh's extension principle for this new framework, expanding fuzzy set theory.

Findings

Generalization of fuzzy numbers to lattice-structured sets

Correction of Zadeh's extension principle for lattices

Application to cognitive maps and expert assessments

Abstract

The concept of a fuzzy number is generalized to the case of a finite carrier set of partially ordered elements, more precisely, a lattice, when a membership function also takes values in a partially ordered set (a lattice). Zadeh's extension principle for determining the degree of membership of a function of fuzzy numbers is corrected for this generalization. An analogue of the concept of mean value is also suggested. The use of partially ordered values in cognitive maps with comparison of expert assessments is considered.