The Decomposition Of Fuzzy Relation And Basis In Semilinear Space

In this paper, we discuss some problems about decomposition offuzzy relation and basis in semilinear space established in residuated lattice. Atfirst, We discuss the decomposition of fuzzy relation in [0, 1], investigate thesquare root problem for the inf ?αcomposite operator, discuss some propertiesof fuzzy matrix which has square roots, then give an algorithm to calculate thesquare roots of a given matrix. Secondly, we study the content of realizable fuzzymatrix. we make use of the necessary and su?cient condition of a fuzzy matrixcan be realized to simplify the algorithm of content for a realizable matrix andgive an easier algorithm which can find the fuzzy matrix B and calculate r(A)within [r(A)]n(n2+ 1)steps based on the work of Wang [Mathematics Annual,(seriesA),6(1999),701-706]; Furthermore, we extend the famous Laplace Theorem tosemiring with the help of bideterminant; At last, we define linear dependence,linear independence baisis and establish semilinear space on residuated latticethen investigate the basis of finite vectors. We got the conclusion that all thefinite vectors have bases which have the same cardinality.