n维模糊集的极小扩展原理

Minimal Extension Principle on N-dimensional Fuzzy Sets and Properties

作为模糊数学理论最基本原理之一,扩展原理具有重要的理论意义与实际应用价值。模糊集的扩展原理主要有两类:极大扩展原理(Zadeh扩展原理)和极小扩展原理,两者在理论和应用中互为补充。因此对n维模糊集极小扩展原理的研究具有重要的意义。根据n维模糊集的截集、分解定理和表现定理,利用模糊集的极小扩展原理,建立n维模糊集的极小扩展原理。首先,对应不同截集下得到的n维模糊集的三个分解定理和三个表现定理,给出n维模糊集极小扩展原理的三种等价表现形式;其次,结合n维模糊集运算的定义及模糊集极小扩展原理的性质,讨论了n维模糊集极小扩展原理的有关性质;最后,给出复合函数的n维模糊集极小扩展原理,并利用复合函数的模糊集极小扩展原理的性质,讨论了复合函数的n维模糊集极小扩展原理的性质。

As the fundamental principle of fuzzy mathematics theory,extension principle possess the important theoretical significance and application value.There are two main kinds of types about extension principle： maximal extension principle and minimal extension principle,which complement each other.The research on minimal extension principle of n-dimensional fuzzy set is therefore significant.According to the cut sets,decomposition theorem and representation theorem of n-dimensional fuzzy set,using the minimal extension principle of fuzzy set,the minimal extension principle of n-dimensional fuzzy set was established.Firstly,in accordance with three kinds of decomposition theorem and three kinds of representation theorem of n-dimensional fuzzy set,three kinds of extension principle of n-dimensional fuzzy set were given.Secondly,under the operations of n-dimensional fuzzy set and the properties of fuzzy set＇s minimal extension principle,the properties of minimal extension principle of n-dimensional fuzzy set were discussed.Finally,the n-dimensional fuzzy set＇s minimal extension principle of the composite functions was also proposed,with their properties acquired.The application of fuzzy set＇s minimal extension principle of the composite functions and its properties was made to have discussed its principle property.