距离分布函数构成的完备格Δ^(+)上的三角模  

Triangular norms on the complete latticeΔ^(+)consisting of distance distribution functions

作　　者：赖洪亮 唐志高 LAI Hong-Liang;TANG Zhi-Gao(School of Mathematics,Sichuan University,Chengdu 610064,China)

机构地区：[1]四川大学数学学院,成都610064

出　　处：《四川大学学报(自然科学版)》2025年第1期25-30,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(12171342)。

摘　　要：在概率度量空间理论中,两个点之间的距离是一个距离分布函数.为了描述概率度量的三角不等式,需要引入距离分布函数构成的完备格Δ^(+)上的三角模.本文基于单位区间[0,1]上的连续三角模T和非负实直线[0,∞]上的连续三角余模S,构造了Δ^(+)上的三角模τ_(T)^(S),并且证明了三角模τ_(T)^(S)与逻辑相关的两个重要性质:(i)τ_(T)^(S)可除当且仅当T=min,S=max;(ii)τ_(T)^(S)满足二次否定律当且仅当T同构于卢卡西维茨三角模,且S同构于单位区间[0,1]上的截断加法.In the theory of probabilistic metric spaces,the distance between two points is a distance distribution function.To describe the triangular inequality of probabilistic metrics,triangular norms onΔ^(+),which is a complete lattice consisting of distance distribution functions,is introduced.By using a continuous triangular norm T on the unit interval[0,1]and a continuous triangular co-norm on the extended non-negative real line[0,∞],one can construct a t-normτ_(T)^(S) onΔ^(+).In this paper,two important properties ofτ_(T)^(S),which are related to logic,is shown:(i)τ_(T)^(S) is divisible if and only if T=min and S=max;(ii)τ_(T)^(S) satisfies the law of double negation if and only if T is isomorphic to theŁukasiewicz triangular norm and S is isomorphic to the truncated addition on[0,1].

关 键 词：距离分布函数 剩余格 三角模 二次否定律 可除 

分 类 号：O153.3[理学—数学]