This work studies the problem of solving a sup-T composite finite fuzzy relation equation, where T is an infinitely distributive strong pseudo-t-norm. A criterion for the equation to have a solution is given. It is proved that if the equation is solvable then its solution set is determined by the greatest solution and a finite number of minimal solutions. A necessary and sufficient condition for the equation to have a unique solution is obtained. Also an algorithm for finding the solution set of the equation is presented.

Inclines are the additively idempotent semirings in which the products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i. e., the matrices over inclines). In this paper, the complete description of the invertible incline matrices is given. Some necessary and sufficient conditions for an incline matrix to be invertible are studied, Cramer's rule over inclines is presented and the group of invertible incline matrices is investigated. The main results in the present paper generalize and develop the corresponding results in the literatures for the Boolean matrices, the fuzzy matrices and the lattice matrices.

on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems 78 (1996) 305-316] and similarity measure of interval valued fuzzy sets, discuss their relationship between similarity measure and entropy of interval valued fuzzy sets in detail, prove three theorems that similarity measure and entropy of interval valued fuzzy sets can be transformed by each other based on their axiomatic definitions and put forward some formulas to calculate entropy and similarity measure of interval valued fuzzy sets.

As every one knows, inverted pendulums are always used to verify some qualities and effect of certain control theory or method. Single or double inverted pendulum control systems are widely used for teaching or researching in universities and some institutes. However how to control a triple inverted pendulum is a well-known difficult problem. Up to now, just a couple of control experts can do the control of a triple inverted pendulum by using some control method. Undoubtedly, to implement controlling of a quadruple inverted pendulum was an unsolved very difficult problem before August 11, 2002 when we first successfully controlled it by using variable universe adaptive fuzzy control method. We demonstrate here how to implement controlling of a quadruple inverted pendulum as well as briefly introducing variable universe adaptive fuzzy control theory.

从解决模糊系统的“规则爆炸”问题出发,本文首先给出广义递阶Mamdani模糊系统的定义,然后证明其与具有中间变量的广义Mamdani模糊系统等价,并借助方形分片线性函数构造性的证明了在最大模和积分模意义下该系统是泛逼近器. 最后仿真实例证实了该系统的有效性.

给出了基于三算法和a三I算法的几种典型模糊系统的插值表达式。指出，基于三I算法和a-三I算法的模糊系统对于某些蕴涵算子具有函数逼近的泛性，而对于不少蕴涵算子只具有阶跃输出能力，而不具有函数逼近的泛性。此外，证明了基于三I算法的模糊系统在一定条件下对于模糊逻辑系统中推理与聚合的次序交换无关。

This paper proves that the maximum order-index of n×n matrices over an arbitrary commutative incline equals (n ? 1)2 + 1. This is an answer to an open problem "Compute the maximum order-index of a member ofMn(L)", proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where Mn(L) is the set of all n × n matrices over an incline L.

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the standard eigenvectors of incline matrices (i. e., matrices over inclines) in detail. As the main results, for an incline matrix with index and for a lattice matrix, the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined. Also it is proved that for an incline matrix with index, the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure. The power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors. The results in the present paper include some previous results for the Boolean matrices, the fuzzy matrices and the lattice matrices among their special cases.

当模糊系统具有插值性时，它必具有泛逼近性。因此，由插值性可以分析模糊系统的逼近能力．本文讨论了由“交”和“并”的方式聚合推理规则所生成的两类模糊系统的插值性问题．首先，通过分析由“单点”模糊化方法、CRI（compositional rule of inference）算法以及“重心法”构造的模糊系统，指出模糊系统是否具有插值性关键取决于模糊蕴含算子的第二个变量为0和1时的表达式或取值。在此基础上，得到两类模糊系统具有插值性的充要条件．最后给出了满足这两个充要条件的一些常用的蕴涵算子。

Purpose-The purpose of this paper is to study the relationship between the property of distributivity and Zadeh's operators. Design/methodology/approach-The results are established by studying the relationship between the condition of distributivity and Zadeh's operations. Findings-When the tconorm T * and t-norm T are Zadeh' s "intersection"and "union" operations, respectively, or when the pseudo-addition % and pseudo-multiplication are Zadeh's "intersection" and "union"operations, respectively, a series of sufficient and necessary conditions for T * and T (or % and) to satisfy a distributivity property are obtained. These results are readily applied to emirings. Originality/value – New theoretical results are discovered and are shown to be useful to the study of triangular operators and applications in intelligent systems.