The K-Quasi-transitive Fuzzy Matrices And The (?)-decomposable Fuzzy Relations

In this paper, the k-quasi-transitive fuzzy matrices and theΘ-decomposable fuzzy relations are discussed.The k-quasi-transitive fuzzymatrix is introduced firstly, the relationship between the k-quasi-transitivefuzzy matrices and other transitive fuzzy matrices are given also,then anecessary and su-cient condition for a fuzzy matrix to be k-quasi-transitive isgiven.The consistency between the k-quasi-transitive fuzzy matrices and theircut matrices is investigated and some properties are shown,the relationshipbetween the k-quasi-transitive fuzzy matrices and the controllable fuzzymatrices is discussed.Next,aΘ-decomposable fuzzy relation on [0,1] isintroduced,namely,let R∈F(X×Y ),if there exist two fuzzy sets A∈F(X)and B∈F(Y ) such that R = AΘB with R(x,y) = (AΘB)(x,y) = A(x)ΘB(y)for all (x,y)∈X×Y ,then R is calledΘ-decomposable, two necessary andsu-cient conditions that a fuzzy relation R isΘ-decomposable are given,thenthe solution set of A and B with R = AΘB is obtained.