基于代数结构的商空间模型研究

现有商空间模型中论域结构一般被指定为拓扑结构，问题的粒度由等价关系唯一地确定。当论域结构由拓扑结构变成应用广泛的代数结构时，引入同余关系的概念，系统地论证了两个重要结论在基于代数结构的商空间模型中依然成立，即全体同余关系构成的完备半序格和保假，保真原理的存在性。而当确定问题粒度的等价关系不是一个同余关系时，对偶地定义了上（下）同余与上（下）商，简捷地证明了它们的存在性并得出了一些重要性质，为商空间的合成与分解提供了理论依据。最后以纠错码进行传输的路由选择算法为实例，分析了基于代数结构的商空间模型在网络安全传输过程中的应用。从结构上扩展了现有商空间模型，为商空间理论与代数理论的结合提供了基础。

A Study for Quotient Space Model Based on Algebraic Structu re

The domain structure in the existing QSM (quotient space model)is usually a topology,and a granule is u-niquely determined by an equivalence relation.However,when the domain structure is assumed as a widely used algebra instead of a topology,it introduces the concept of congruence relation,and systematically demonstrates the existence of two basic conclusions in QSM based on algebraic structure—all the congruence relations forming a complete semi-order lattice and the principles of fal-sity preserving and truth preserving.And when the equivalence relation determining a granule is not a congruence relation,it de-fines the concepts of least upper (greatest lower)congruence and least upper (greatest lower)quotient in antithesis,proves their existence for simplicity,and discusses some of their important properties which are the theoretical basis for the composition and decomposition of different granularities.Finally,based on the routing algorithm transmitting as error correcting codes,it analyzes the application of QSM based on algebraic structure during network secure transmission.The paper extends the theory of QSM from structure,and provides theoretical basis for the combination of quotient space theory and algebraic theory.