摘要
本文的主要结论是:一切Fuzzy格皆由极小分支构成,极小分支则由素元组成。具体地说,本文以正交集为工具,引入了分支,极大、极小分支的概念。证明了极小分支的每个非零元为素元。属于同一极小分支的各素元彼此都不正交,而属于不同极小分支里的素元彼此一定正交,每个Fuzzy格必存在极小分支的完全族。Fuzzy格的每个元素都可向极小分支投影,并且等于各投影的并。即Fuzzy格的各个元可以一意地表成彼此正交的若干素元的并。特别,对于正统Fuzzy格,每个分支的最大元必为分明元,反之每个分明元的下集必是一个分支。分明元与分支一一对应。极小分明元对应于极小分支。在极小分支里,由原来的补结构,可以诱导建立极小分支里的补结构。这时的极小分支也成为一个Fuzzy格(它只有一个分支)。当且仅当这些极小分支Fuzzy格彼此同构时,原Fuzzy格才与L-Fuzzy集同构(即Fuzzy格退化为L-Fuzzy集);当且仅当每个极小分支由一个元素组成时,正统Fuzzy格退化为Boolean代数(格)。非退化的正统Fuzzy格的主要特征在于它有非分明元。正统Fuzzy格不同于L-fuzzy集主要是它的极小分支可以不同构。
The main results of this paper are following: A fuzzy lattice is formed of the minimal components and every minimal component is constituted by the prime elements. It was shown that every non-zero element of the minimal component is a prime one. Any two elements are non-disjoint in a minimal component. If two elements are belong to two different minimal component respectively, then they are disjoint each other. A prime element belongs to and only belong to a minimal component. Complete family of the minimal components certainly exists in a fuzzy lattice. Any element of the fuzzy lattice has a corresponding projection on every minimal component, and equal to their union. In other words, every element of the fuzzy lattice can be definitely represented as an union of the primes which are disjoint each other. Especially, for orthodox fuzzy lattice the maximal element of any component is a crisp element, and the down set of any crisp element is a component. Both the component and the crisp element are corresponding one to one each other, and the minimal component corresponds to the minimal crisp element. In the minimal component of the orthodox fuzzy lattice, we can build complementary stucture by means of primary complementary stucture. In this time, a minimal component has become a fuzzy lattice. If and only if these minimal components (fuzzy lattice) are isomorphic each other, original fuzzy lattice is isomorphic to a L-fuzzy set. If and only if every minimal component is constituted by unique element, the orthodox fuzzy lattice is degenerate to Boolean lattice.
出处
《模糊系统与数学》
CSCD
1989年第1期56-64,共9页
Fuzzy Systems and Mathematics
