连续并既约元及其在刻画Fuzzy关系方程解集中的应用

Continuous Join-Irreducible Elements and Their Applications to Describing the Solution Set of Fuzzy Relational Equations

本文首先引入连续并既约元(是并既约元但不是完全并既约元的元)的概念,并讨论了它的性质,然后应用连续并既约元的性质去刻画完备Brouwer格上无限Fuzzy关系方程A☉X=b的解集(其中A=(aj)j∈J和b已知,b为连续并既约元,X= (xj)j∈JT未知,“☉”表示“sup-inf”,J为无限集):给出了方程存在可达解与不可达解的充要条件及可达解与不可达解的一些性质,进一步刻画了方程的解集．

In this paper, we first introduce a concept of continuous join-irreducible element （which is join-irreducible but not completely join-irreducible） in lattices and discuss some of its properties. Then we use the properties to describe the solution set of fuzzy relational equation A ⊙ X = b （where both A = （aj）j∈J and b are known, b is a continuous join-irreducible element, X = （xj）j∈J^T is unknown, ＂⊙＂ represents ＂supinf＂, J is an infinite set） on complete Brouwerian lattices. First, we give a necessary and sufficient condition for the existence of attainable solutions （resp. unattainable solutions） and some properties of attainable solutions （resp. unattainable solutions）. Then we describe further the solution set of fuzzy relational equations.