理想生成的格上粗集及格值粗集的研究

发布时间：2018-04-17 21:54

本文选题：粗集 + CCD格　；参考：《湖南大学》2016年博士论文

【摘要】：粗集模型的扩展是粗集理论研究的一个重要内容.利用代数系统来推广粗集理论是一个研究的重点.2006年,陈等将完备的完全分配格(简称CCD格)引入到粗集理论中作为基本代数系统,在CCD格上定义了覆盖,并通过该覆盖定义了更为一般和抽象的近似算子.2013年,基于CCD格上的覆盖诱导的邻域,秦等又定义了一种下近似算子和三种上近似算子,并讨论了它们与陈等提出的覆盖近似算子之间的关系.另一方面,周和胡于2014年在CCD格上定义了关系,并通过关系构造了下和上近似算子.本文在此基础上,借助于CCD格中的理想,分别从关系和覆盖诱导的邻域两个方面定义和研究了CCD格上新的近似算子.在第二章中,基于CCD格L上的二元关系,我们通过L中的理想I定义了一对新的近似算子,其可以看作是周和胡的近似算子的推广.当I是L的最小理想且R是自反的二元关系时,上述两种粗近似一致.当L是完备的原子布尔格且R是自反和传递的二元关系时,我们给出了上述两种粗近似一致的等价刻画.证明了新的近似算子对元素的逼近度更高,并通过例子解释了这样的结果.此外,讨论了新的近似算子的拓扑和格结构.在第三章中,基于CCD格L上的覆盖诱导的邻域,我们利用L中的理想定义了新的覆盖近似算子.新的近似算子是秦等引入的近似算子的推广,而且对CCD格中的元素的逼近度更高.当L是幂集格时,我们定义的近似算子恰好是基于adhesion的覆盖近似算子.讨论了新的近似算子与CCD格上已有的覆盖近似算子之间的关系.第四章进一步研究了φ-模糊粗集.φ-模糊粗集可以看成是我们熟知的R-模糊粗集的一种推广.首先利用截集的思想定义了一族L-模糊关系,由此给出了φ-模糊粗集的表示定理,即φ-模糊粗集可以由一族R-模糊粗集来表示.其次,得到了映射φ一个更为直观的意义,即把φ解释为普通的邻域算子在L-模糊情形下的推广.讨论了特殊类的φ下相应的φ-模糊粗集.最后,不是去限制格L,而是通过约束映射φ,给出了φ-模糊粗集诱导的L-拓扑.第五章讨论了Quantale中的格值理想.考虑格值为完备剩余格,从模糊点和截集等方面,给出了Quantale中的格值理想的刻画.我们着重利用格值模糊集诱导的R-模糊粗集刻画了Quantale中的格值理想.此外,定义了Quantale中的格值拟粗理想,讨论了它与格值理想和格值粗理想之间的关系.
[Abstract]:The extension of rough set model is an important part of rough set theory research.In 2006, Chen et al introduced complete complete distributive lattices (CCD lattices) into rough set theory as basic algebraic systems, and defined covering on CCD lattices.In 2013, based on the overlay induced neighborhood on CCD lattice, Qin et al. defined a lower approximation operator and three upper approximation operators.The relation between them and the covering approximation operator proposed by Chen et al is discussed.On the other hand, Zhou he Hu defined the relation on the CCD lattice in 2014, and constructed the lower and upper approximation operator through the relation.In this paper, based on the ideals in CCD lattices, we define and study new approximate operators on CCD lattices from two aspects of relation and covering induced neighborhood, respectively.In the second chapter, based on the binary relation on CCD lattice L, we define a pair of new approximate operators by the ideal I in L, which can be regarded as a generalization of Zhou Hehu's approximation operator.When I is the minimum ideal of L and R is a reflexive binary relation, the two rough approximations are consistent.When L is a complete atomic Boolean lattice and R is a binary relation between reflexive and transitive, we give the equivalent characterizations of the two rough approximations mentioned above.It is proved that the new approximation operator has a higher degree of approximation to elements, and this result is explained by an example.In addition, the topology and lattice structure of new approximate operators are discussed.In chapter 3, we define a new covering approximation operator based on the covering induced neighborhood of CCD lattice L.The new approximation operator is a generalization of the approximation operator introduced by Qin et al. and the approximation degree of the elements in the CCD lattice is higher.When L is a power set lattice, the approximation operator we define happens to be a covering approximation operator based on adhesion.The relation between the new approximation operator and the covering approximation operator on CCD lattice is discussed.In chapter 4, we further study 蠁 -fuzzy rough sets. 蠁 -fuzzy rough sets can be regarded as a generalization of R-fuzzy rough sets.Firstly, a family of L-fuzzy relations is defined by using the idea of cut sets, and the representation theorem of 蠁 -fuzzy rough sets is given, that is, 蠁 -fuzzy rough sets can be represented by a family of R-fuzzy rough sets.Secondly, a more intuitionistic meaning of mapping 蠁 is obtained, that is, 蠁 is interpreted as a generalization of ordinary neighborhood operators in the case of L- fuzzy.The corresponding 蠁 -fuzzy rough sets under 蠁 for special classes are discussed.Finally, instead of limiting lattice L, we give a 蠁 -fuzzy rough set induced L-topology by constrained mapping 蠁.In chapter 5, we discuss the lattice ideal in Quantale.Considering that the lattice value is a complete residual lattice, the characterizations of lattice valued ideals in Quantale are given in terms of fuzzy points and cut sets.In this paper, lattice valued ideals in Quantale are characterized by R- fuzzy rough sets induced by lattice-valued fuzzy sets.In addition, the lattice-valued quasi-rough ideals in Quantale are defined, and the relations between lattice valued ideals and lattice-valued rough ideals are discussed.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177;O159

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