Minimal Solutions Of Fuzzy Relation Equations Over Complete Brouwerian Lattices And Some Matrix Decomposition Problems

In this paper, we consider minimal solutions of fuzzy relation equations over complete Brouwerian lattices and some matrix decomposition problems. First, concepts of a minimal common join decomposition and a basic minimal common join decomposition are introduced in complete lattices, then the existence conditions, characterizations and properties of such decompositions are discussed. As their applications, computations of minimal solutions of a fuzzy relation equation are given over complete Brouwerian lattices. In particular, the number of minimal solutions of a fuzzy relation equation over complete lattices under some conditions is given. Sec-ond, we give a formula for content of 3-order nonnegative integral symmetric realizable matrices, show several necessary and sufficient conditions for the realizability of n-order nonnegative integral symmetric matrices, obtain some algorithms which not only determine whether a matrix is realizable but also give one of its realization matrices and content when it is realizable. Third, interpreting Boolean matrices as undirected graphs, we prove that the con-tent of a realizable Boolean matrix is just the sum of the clique cover number of its corresponding undirected graph and the number of isolated vertices in this graph. At last, we present a necessary and sufficient condition for the existence of a square root for a given matrix based on sup-T composition, and give a theoretical algorithm to compute all square roots of a given matrix.