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Algebraic structure of the Gaussian-PDMF space and applications on fuzzy equations

Chuang Zheng

TL;DR

It is demonstrated that the Gaussian-PDMF space exhibits a well-defined algebraic structure, featuring a subspace that forms a division ring, allowing for the representation of fuzzy polynomials, among other properties.

Abstract

In this paper, we extend the research presented in [Wang and Zheng, Fuzzy Sets and Systems, p108581, 2023] by establishing the algebraic structure of the Gaussian Probability Density Membership Function (Gaussian-PDMF) space. We consider fixed objective and subjective entities, denoted as $(h,p)$, and provide the explicit form of the membership function. Consequently, every fuzzy number with the membership function in $X_{h,p}(\mathbb{R})$, denoted as $\tilde{x}$, can be uniquely identified by a vector $\langle x; d^-, d^+, μ^-,μ^+\rangle$. Here, $x\in \mathbb{R}$ represents the "leading factor" of the fuzzy number $\tilde{x}$ with a membership degree equal to $1$. The parameters $d^-$ (left side) and $d^+$ (right side) denote the lengths of the compact support, while $μ^-$ (left side) and $μ^+$ (right side) represent the shapes. We introduce five operators: addition, subtraction, multiplication, scalar multiplication, and division. We demonstrate that, based on our definitions, the Gaussian-PDMF space exhibits a well-defined algebraic structure. For instance, $X_{h,p}(\mathbb{R})$ is a vector space over $\mathbb{R}$, featuring a subspace that forms a division ring, allowing for the representation of fuzzy polynomials, among other properties. We provide several examples to illustrate our theoretical results.

Algebraic structure of the Gaussian-PDMF space and applications on fuzzy equations

TL;DR

It is demonstrated that the Gaussian-PDMF space exhibits a well-defined algebraic structure, featuring a subspace that forms a division ring, allowing for the representation of fuzzy polynomials, among other properties.

Abstract

In this paper, we extend the research presented in [Wang and Zheng, Fuzzy Sets and Systems, p108581, 2023] by establishing the algebraic structure of the Gaussian Probability Density Membership Function (Gaussian-PDMF) space. We consider fixed objective and subjective entities, denoted as , and provide the explicit form of the membership function. Consequently, every fuzzy number with the membership function in , denoted as , can be uniquely identified by a vector . Here, represents the "leading factor" of the fuzzy number with a membership degree equal to . The parameters (left side) and (right side) denote the lengths of the compact support, while (left side) and (right side) represent the shapes. We introduce five operators: addition, subtraction, multiplication, scalar multiplication, and division. We demonstrate that, based on our definitions, the Gaussian-PDMF space exhibits a well-defined algebraic structure. For instance, is a vector space over , featuring a subspace that forms a division ring, allowing for the representation of fuzzy polynomials, among other properties. We provide several examples to illustrate our theoretical results.
Paper Structure (6 sections, 12 theorems, 53 equations, 5 figures)

This paper contains 6 sections, 12 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Arithmetic operators on $X_{h,p}(\mathbb{R})$
  4. Vector space
  5. Numerical simulation
  6. Final remarks

Key Result

Theorem 2.1

Let and $h(x)$ is continuous and increasing on $(0,1)$. Let $p^-$ and $p^+$ be originated from the same class $p$ of PDFs. Then there exists at least one pair $(h, p)$ such that the graph of $f_{h,p}$ passes through $P$ and $Q$, i.e., $f_{h,p}(x^-)=y^-$ and $f_{h,p}(x^+)=y^+$. Moreover, the PDMF in the fulfills all requirements in Definition 1st_Def and Definition 2nd_Def.

Figures (5)

  • Figure 1: Fuzzy number $\tilde{1}=\langle(-1,1,2);P,Q\rangle$ or $\langle 2; 2,1,\mu^{-},\mu^{+}\rangle$
  • Figure 2: The graph of $\tilde{3}$ and $\tilde{1}$
  • Figure 3: The graph of $\frac{1}{2}(\tilde{3}\ominus\tilde{1})$
  • Figure 4: The graph of $\tilde{2}$
  • Figure 5: The graph of $\tilde{2}^{-1}\otimes (\tilde{3}\ominus\tilde{1})$

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.1
  • Remark 3.4
  • Proposition 3.1
  • ...and 15 more