Conditions for lossless join

There is a well-known algorithm for determining when a decomposition ˆ {R1,R2, . . . ,Rm} of a database relation scheme has a lossless join with respect to a set of functional dependencies. We ®rst present a reformulation of this algorithm in terms of set closures. For the special case of mˆ 2, there is a well-known explicit condition for losslessness. Our formulation extends this result for general m. Also, our formulation leads to a strong necessary condition for to be lossless. Separately, we prove a sucient condition for to be lossless. Finally, we present a sucient condition, and a necessary condition for to be lossless with respect to a set of functional and multivalued dependencies.