Mathematics

CLC number: O159

The Structure and Properties of (ξ, s) - Generated Implications

-生成蕴涵的构造及其性质

¹ School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, Hubei, China (湖北民族大学　数学与统计学院，湖北　恩施　445000)
² School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, Shaanxi, China (陕西师范大学　数学与统计学院，陕西　西安　710119)

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 20 April 2025

Abstract

Fuzzy implication, as a critical logical connective, holds substantial research significance for its construction and characterization within the domain of fuzzy logic. In this paper, we first construct a novel class of fuzzy implications, termed $(\xi ,s)$-generated implications, by employing a pair of multiplicative and additive generators derived from triangular conorms. Secondly, we investigate the fundamental properties of this new class of implications. Finally, we examine the relationships between the $(\xi ,s)$-generated implications and other established classes of fuzzy implications, including $\mathrm{Y}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{r}$-, $h$-, $k$-, $(\theta ,t)$-, $(S,N)$- and $R$-implication. The results indicate that the $(\xi ,s)$-generated implications distinctly differ from these known types.

摘要

模糊蕴涵作为重要的逻辑连接词，对其构造和刻画都有着深刻的理论研究意义。本文首先通过三角余模的一对乘法和加法生成子，构造了一类新的模糊蕴涵，称为$(\xi ,s)$-生成蕴涵。其次，研究了这类蕴涵的基本性质。最后，讨论了$(\xi ,s)$-生成蕴涵与其他类模糊蕴涵的关系，结果表明，$(\xi ,s)$-生成蕴涵不同于已知的$\mathrm{Y}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{r}$-、$h$-、$k$-、$(\theta ,t)$-、$(S,N)$-以及$R$-等蕴涵。