The Solutions Of A Fuzzy Relation Equation In A Complete Brouwer Lattice And The Problem On The Number Of Transitive Relations

This paper discusses the problem of resolution of a fuzzy relation equation defined on a complete Brouwer lattice,the number of transitive Boolean matrices of order n and the convergence of fuzzy matrices.First, in the case of finite domains,it is shown that there exists an element∑∈(?) (where (?) is the solution set of a fuzzy relation equation) such that [X∧∑)∈(?) for every element X∈(?) and∑is the fuzzy union of all minimal elements in (?) if (?)≠φand every component of B with an irredundant finite join-decomposition.Then we use some related graph theory to give the upper and lower bounds of the number of transitive Boolean matrices of order n.from the structure of Boolean matrices.Finally, we stress the significance of the group-theoretical approach and show many results about the convergence of matrices can be obtained in an unified way. And the convergence of powers of the multiplication of two matrices over distributive lattices are also investigated.