Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques

TL;DR

This paper develops a combinatorial formula to count non-equivalent fuzzy matrices of finite order, introducing a new sequence and analyzing their properties under an equivalence relation.

Contribution

It presents an explicit combinatorial formula for counting distinct fuzzy matrices, introduces a new sequence, and characterizes properties of equivalence classes of fuzzy matrices.

Findings

Derived a new combinatorial sequence for fuzzy matrices

Characterized properties of non-equivalent fuzzy matrix classes

Enumerated the number of distinct fuzzy matrices of a given order

Abstract

The novelty of this paper is to construct the explicit combinatorial formula for the number of all distinct fuzzy matrices of finite order, which leads us to invent a new sequence. In order to achieve this new sequence, we analyze the behavioral study of equivalence classes on the set of all fuzzy matrices of a given order under a suitable natural equivalence relation. In addition this paper characterizes the properties of non-equivalent classes of fuzzy matrices of order n with elements having degrees of membership values anywhere in the closed unit interval [0,1]. Further, this paper also derives some important relevant results by enumerating the number of all distinct fuzzy matrices of a given order in general. And also, we achieve these results by incorporating the notion of k-level fuzzy matrices, chains, and flags (maximal chains). Keywords: Fuzzy matrices; k-level fuzzy matrices; Chains; Flags; Binomial numbers