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

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