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On the solvability of bipolar max-product fuzzy relation equations with the product negation

M. Eugenia Cornejo, David Lobo, Jesús Medina

Abstract

This paper studies the solvability of the max-product fuzzy relation equations in which a negation operator is considered. Specifically, the residuated negation of the product t-norm has been introduced in these equations in order to increase the flexibility of the standard fuzzy relation equations introduced by Sanchez in 1976. The solvability and the set of solutions of these bipolar equations have been studied in different scenarios, depending on the considered number of variables and equations.

On the solvability of bipolar max-product fuzzy relation equations with the product negation

Abstract

This paper studies the solvability of the max-product fuzzy relation equations in which a negation operator is considered. Specifically, the residuated negation of the product t-norm has been introduced in these equations in order to increase the flexibility of the standard fuzzy relation equations introduced by Sanchez in 1976. The solvability and the set of solutions of these bipolar equations have been studied in different scenarios, depending on the considered number of variables and equations.

Paper Structure

This paper contains 8 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Bipolar max-product FREs with the product negation
  4. Solving bipolar max-product FREs with one unknown variable
  5. Solving bipolar max-product FREs with different unknown variables
  6. Computing maximal and minimal solutions
  7. Solving bipolar max-product FRE systems
  8. Conclusions and future work

Key Result

Theorem 1

Let $f\colon[a,b]\rightarrow \mathbb{R}$ be a continuous function and $c\in\mathbb{R}$ be any number between $f(a)$ and $f(b)$ inclusive. Then, there exists $x\in[a,b]$ such that $f(x)=c$.

Theorems & Definitions (21)

  • definition 1
  • definition 2
  • definition 3
  • Theorem 1: Intermediate Value Theorem
  • definition 4
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 11 more