一种新的模糊广义拟阵

A new fuzzy greedoids

基于Al-Hawary所提出的模糊广义拟阵(亦称(H)模糊广义拟阵),讨论(H)模糊广义拟阵性质的不足,提出新的模糊广义拟阵,并研究其相关性质.首先否定Al-Hawar所给出的结论:(H)模糊广义可行集族((亦称(H)模糊可行系统)的r-水平截集族仍然是清晰可行集组成的集合,其中r∈(0,1].其次,重新研究(H)模糊可行系统的r-水平截集族,得到了(H)模糊可行系统的相关性质.进而,对(H)模糊广义拟阵的不足进行分析,提出了一种新的模糊广义拟阵(亦称为(Y)模糊广义拟阵),并研究了其性质,得到了(Y)模糊可行系统(即(Y)模糊广义可行集族)与某一清晰可行系统(即清晰可行集族,由(Y)模糊广义可行集的支集构成的集合)等价的结论,这一结论表明(Y)模糊广义拟阵在一定程度上优越于(H)模糊广义拟阵.最后通过有向和无向图分别构造有向分支和无向分支(Y)模糊广义拟阵;还根据Hasse图构造了偏序(Y)模糊广义拟阵.

Based on the concept of fuzzy greedoid, which is proposed by A1-Hawary and nanled as （H） fuzzy greedoids in this paper,the purpose of this work is to discuss the deficiency of （H） fuzzy greedoids, to present a new fuzzy greedoid,and to study some properties related to the new fuzzy greedoid. Firstly,we do not agree with A1-Hawary＇s conclusion, which shows that a family of all r-cut sets of fuzzy feasible sets is a family of a crisp feasible sets, where r E （0,1], and we verify that this conclusion is incorrect through several examples, which are given under different conditions. Secondly,we restudy the family of all r-cut sets of （H） fuzzy feasible system,and get some properties as- sociated with （H） fuzzy feasible system. Thirdly,after analyzing the deficiency of （H） fuzzy greedoids,we present a new fuzzy greedoid, which is called a （Y） fuzzy greedoid, and study some properties of the new concept. More specifically,We prove that （Y） fuzzy greedoids must be （H） fuzzy greedoids, but on the contrary, we cannot conclude that （H） fuzzy greedoids are （Y） fuzzy greedoids,and this conclusion is verified through an example. We also obtain some properties about the family of all r-cut sets of （Y） fuzzy feasible system. Most importantly, by discussing properties of this new fuzzy greedoid, we obtain that （Y） fuzzy feasible system is equivalent to a crisp feasible system,which is a set consisting of support sets of （Y） fuzzy feasible sets,and this conclusion demonstrates that （Y） fuzzy greedoids are superior to （H） fuzzy greedoids in a certain degree. Finally,we discuss the construction of （Y） fuzzy greedoids based on different types of graph. Specifically, we present methods to get a directed and undirected branching （Y） fuzzy greedoid based on a directed and undirected graph, respectively. Similarly, we present another method to construct a poset （Y） fuzzy greedoid based on a Hasse graph. Moreover,two examples are given to show the construction of an undirected branching （Y） fuzzy greedoid and a poset （Y） fuzzy greedoid by using our methods, respectively.