伪效应代数与序列效应代数范畴性质的研究

项目名称： 伪效应代数与序列效应代数范畴性质的研究

项目编号： No.11201278

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 郭建胜

作者单位： 陕西师范大学

项目金额： 22万元

中文摘要： 效应代数是Hilbert空间效应的抽象，序列效应代数是带有序列积的效应代数，它们是量子逻辑中两个重要的模型。本项目主要研究以下三方面内容。（1）用范畴学的思想和方法研究量子逻辑：建立效应代数（伪效应代数）范畴中的各种概念，然后具体找出效应代数（伪效应代数）中投射元、内射元、等子、余等子、极限、余极限、态射等结构以及这些结构之间的联系和等价刻画。（2）用代数的思想和方法研究量子逻辑：具体构造出效应代数（伪效应代数）中的理想、滤子、同余以及模糊理想、模糊滤子和模糊同余等代数结构以及由模糊同余所确定的商代数，并研究模糊理想、模糊滤子和模糊同余之间是否存在一一对应关系，从而得到效应代数的清晰的代数结构及性质。（3）研究序列效应代数的理想、同余：给出序列效应代数中的理想、滤子和同余的概念及性质；研究序列效应代数的商、范畴表示、范畴等价等。

中文关键词： 效应代数；伪bck代数；理想；软集；粗糙集

英文摘要： Effect algebra is an abstract algebra of Hilbert space effects and sequential effect algebra is a special effect algebra with a sequential product. They are very important models in quantum logic research. The main content in this project are following. (1) Studying the quantum logic with the ideals and methodes in category: we will give the all kinds of concepts in effect algebra category,then we will give the concrete structures of projective objects, injective objects, equalizers, coequalizers, limits, colimits and morphisms, and give the relations and equivalent descriptions in all concepts. (2) Studying the quantum logic with the ideals and methodes in algebra: we will give the concrete structures of ideals,filters,congruences,fuzzy ideals,fuzzy filters and fuzzy congruences in effect algebras(pseudo effect algebras) and the quotient algebras decided by fuzzy congruences, and we will study whether there are one to one relations in fuzzy ideals,fuzzy filters and fuzzy congruences. Sequentially,we will get the clear algebraic structures and properties of effect algebras. (3)Studying the ideals and congruences of sequential effect algebras: we will give the concepts and properties of ideals,filters and congruences in sequential effect algebra;we will research the quotient ,category representation and category

英文关键词： Effect algebra；Pseudo-bck algebra；Ideal；Soft set；Rough set