关于二型模糊集合的一些基本问题

采用集合论的方法给出了单位模糊集合和二型模糊集合及其在一点的限制等定义，使得二型模糊集合更易于理解.通过定义嵌入单位模糊集合来描述一般二型模糊集合，并给出离散、半连通二型模糊集合的表达式.根据论域、主隶属度及隶属函数的特性将二型模糊集合分为四种类型:离散、半连通、连通及复合型，并根据连通的特点将连通二型模糊集合分为单连通及多连通两类.利用支集的闭包（Closure of support，CoS）划分法表述主隶属度及区间二型模糊集合.提出了CoS二、三次划分法分别来表述单、复连通二型模糊集合，并使每一个子区域的上下边界及次隶属函数在该子区域上的限制分别具有相同的解析表述式.最后，探讨了二型模糊集合在一点的限制、主隶属度、支集、嵌入单位模糊集合之间的关系.

Some Fundamental Issues on Type-2 Fuzzy Sets

New and revised definitions such as unit fuzzy sets, type-2 fuzzy sets (T2 FSs) and their restriction at a point are introduced by the method of set theory to make them easily understandable. All the definitions are uniform and formal, and able to represent type-2 fuzzy sets conveniently. An embedded unit fuzzy set is defined to describe general T2 FSs and the representations of discrete and partially connected T2 FSs are given as well. Based on the universes of discourse, primary membership grade, and membership function, type-2 fuzzy sets are divided into four classes:discrete, partially connected, connected, and compounded. Connected T2 FSs are further classified into two types:single connected and multi-connected. The partition method for closure of support (CoS) is used to represent the primary membership grades and interval type-2 fuzzy sets. A method of second/third partition for CoS is proposed to represent single connected/multi-connected T2 FS, respectively after CoS is divided twice and three times; the upper and lower membership functions, and the restriction of the secondary membership function have same analytic expressions, respectively. Finally, the connections among the restriction of T2 FS at a point, primary membership function, CoS, embedded type-1 fuzzy set are discussed.