When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values?

TL;DR

This paper investigates the conditions under which convolution-based operators form t-norms on specific sets of fuzzy truth values, impacting the design of rule-based fuzzy systems.

Contribution

It provides necessary and sufficient conditions for convolution operators to be t-norms on convex, normal, upper semicontinuous fuzzy truth values.

Findings

Characterizes when convolution operators are t-norms on the set of convex, normal, upper semicontinuous fuzzy truth values.

Extends previous work by addressing the case of upper semicontinuous fuzzy truth values.

Offers criteria to select suitable t-norms for fuzzy rule inference systems.

Abstract

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 $(\mathbf{L_u}, \sqsubseteq)$.