源于非经典逻辑的代数结构研究综述  

A Survey of Algebraic Structures Derived from Non-classical Logics

作　　者：张小红[1] ZHANG Xiaohong(Research Center for Data Intelligence and Threeway Decision,Shaanxi University of Science and Technology,Xi’an 710072,Shaanxi)

机构地区：[1]陕西科技大学中加数据智能与三支决策研究中心,陕西西安710072

出　　处：《四川师范大学学报（自然科学版）》2019年第1期1-14,共14页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(61573240;61473239);陕西省教育厅2018年度专项科学研究计划(18JK0099)

摘　　要：作为智能科学的数学基础之一,非经典逻辑(主要指非经典数理逻辑)及相关代数结构扮演着十分重要的角色.对源于非经典逻辑的代数结构进行全面系统总结,从蕴涵片段的视角梳理其中的内在联系,这些代数结构包括各种可换剩余格、非可换剩余格、非结合剩余格、剩余有序广群、BCK/BCI-代数、BCC/BZ-代数、伪BCK/BCI-代数等.同时介绍近年来非经典逻辑代数方向的最新研究进展,包括量子B-代数(quantum B-agebra)、EO-代数(extended-order algebra)及新近提出的基本蕴涵代数(basic implication algebra)等.As one of the mathematical foundations of intelligent science,non-classical logic(mainly referring to non-classical mathematical logic)and related algebraic structures play an important role.In this paper,the algebraic structures derived from non-classical logics are summarized comprehensively and systematically,and their internal relations are sorted out from the perspective of implication fragments.These algebraic structures include various commutative residual lattices,non-commutative residual lattices,non-associative residual lattices,residual ordered groupoids,BCK/BCI-algebras,BCC/BZ-algebras,pseudo BCK/BCI-algebras,etc.At the same time,this paper introduced the latest research progress of non-classical logic algebras in recent years,including quantum B-algebras,EO-algebras and basic implication algebras.

关 键 词：非经典逻辑 模糊逻辑 代数结构 蕴涵代数 滤子 

分 类 号：O141[理学—数学] O159[理学—基础数学]