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

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 $⟨x;d^{-},d^{+},{\mu }^{-},{\mu }^{+}⟩$. 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 ${\mu }^{-}$ (left side) and ${\mu }^{+}$ (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.