Lossless join decomposition

In database design, a lossless join decomposition is a decomposition of a relation ${\displaystyle r}$ into relations ${\displaystyle r_{1},r_{2}}$ such that a natural join of the two smaller relations yields back the original relation. This is central in removing redundancy safely from databases while preserving the original data.^[1] Lossless join can also be called non-additive.^[2]