When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values?
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| Format: | Preprint |
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2026
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| _version_ | 1866908798957912064 |
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| author | Sun, Jie |
| author_facet | Sun, Jie |
| contents | In Type-2 rule-based fuzzy systems (T2 RFSs), triangular norms on complete lattice $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ can be used to model the compositional rule of inference, where $\textbf{L}$ is the set of all convex normal fuzzy truth values, $\mathbf{L_u}$ is the set of all convex normal and upper semicontinuous fuzzy truth values, and $\sqsubseteq$ is the so-called convolution order. Hence, the choice of t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ may influence the performance of T2 RFSs, and thus, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$, the mainstream method is based on convolution $\ast_\vartriangle$ induced by two operators $\ast$ and $\vartriangle$ on the unit interval $[0,1]$. Recently, we have complete solve the question when convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L},\sqsubseteq)$. This paper aim to provide the necessary and sufficient conditions under which convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L_u}, \sqsubseteq)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values? Sun, Jie General Mathematics In Type-2 rule-based fuzzy systems (T2 RFSs), triangular norms on complete lattice $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ can be used to model the compositional rule of inference, where $\textbf{L}$ is the set of all convex normal fuzzy truth values, $\mathbf{L_u}$ is the set of all convex normal and upper semicontinuous fuzzy truth values, and $\sqsubseteq$ is the so-called convolution order. Hence, the choice of t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ may influence the performance of T2 RFSs, and thus, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$, the mainstream method is based on convolution $\ast_\vartriangle$ induced by two operators $\ast$ and $\vartriangle$ on the unit interval $[0,1]$. Recently, we have complete solve the question when convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L},\sqsubseteq)$. This paper aim to provide the necessary and sufficient conditions under which convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L_u}, \sqsubseteq)$. |
| title | When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values? |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2601.22190 |