若干逻辑代数上超运算符的研究

本项目拟舍弃运算结果唯一性的要求，在效应代数、格、BCK-代数中将特殊子类性质中的子集的最小元与最大元分别用全体极小元集与全体极大元集来代替，在更一般的子类上定义超交运算符、超并运算符、与积运算相伴随的超蕴涵算子、与蕴涵算子相伴随的超积运算符等各类与序关系紧密相关的超运算符。这一类超运算符可以更加直观的展现逻辑代数的偏序结构，揭示逻辑代数中偏序结构与代数结构的联系。借助这些超运算符，我们可以扩大剩余理论的应用范围，将逻辑代数中一些特殊子类的性质推广到更一般的类中，进而发掘这些逻辑代数的内在联系。本项目的研究可以为其它逻辑代数的研究提供新的思路和方法，为超代数系统的理论提供背景和素材。

The project is intended to define hyper meet operations, hyper join operations, hyper residual implications, hyper product operations and some other hyper operation which has closely relation with order structure by replacing the least element and the greatest element in the properties of some special class of effect algebras, lattice, BCK-algebras by the set of all minimal upper bounds and the set of all maximal lower bounds of a set, respectively, which give up the requirement that the result of an operation must only contain no more than one element. These hyper operations can show the order structure of some logic algebras directly, and reveal the relationship between the order structure and the algebraic structure of a logic algebra. With the help of these hyper operations, some results on special subclass of a logic algebra may extend to a wider class. Moreover, hyper structures of logic algebras can be explored through hyper operations. This project can offer various approaches to some other logical algebras and provide references and material for the research of hyper algebras.