Structure of multivalued dependencies in relation scheme

Abstract

Abstract. As pointed out in [1],Multivalued dependencies (MVD) depend on the context in which they are defined and thus are very hard to visualize. In this paper, continuing the study in [1], we give some results which may provide some more insight regarding the existence of possible MVDs in the relation schemes provided all their FDs are known in advance.Throughout this paper, we assume that the reader is familiar with the basic concepts of the relational database [2,3]. MULTIVALUED DEPENDENCIESLet U be a set attributes, X and Y be disjoint subsets of U. We say that in the relation R over the set of attributes U, there is a multivalued dependency of the set Y Let R be a relation scheme, an MVD X --+-+ Y is said to be nontrivial if Y =1= 0, Y~X and XY =1= U. Since X -+-+ Y is valid iff X --+-+ Y \ X is valid, we shall always assume that left and right sides of an MVD are disjoint.Let Z = U \ XY, since X -+-+ Y holds iff X -+-+ Z holds, we often write the MVD X --+-+ Y as X -+-+ YIZ. If the FD X -+ Y or the FD X -+ Z is valid, then the MVD X -+-+ YIZ also holds in R. In this case, we call the MVD X -+--+ YIZ, an MVD counterpart of an FD. We call an MVD X -+-+ YIZ pure if it is nontrivial and it is not an MVD counterpart of an FD inR.In [1] is has been shown that: