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]

Criteria

Lossless join can also be called nonadditive.^[2]

If ${\displaystyle R}$ is split into ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$, for this decomposition to be lossless (i.e., ${\displaystyle R_{1}\bowtie R_{2}=R}$) then at least one of the two following criteria should be met.

Examples

• Let ${\displaystyle R=(A,B,C,D)}$ be the relation schema, with attributes A, B, C and D.
• Let ${\displaystyle F=\{A\rightarrow BC\}}$ be the set of functional dependencies.
• Decomposition into ${\displaystyle R_{1}=(A,B,C)}$ and ${\displaystyle R_{2}=(A,D)}$ is lossless under F because ${\displaystyle R_{1}\cap R_{2}=A)}$. A is a superkey in ${\displaystyle R_{1}}$, meaning we have a functional dependency ${\displaystyle \{A\rightarrow BC\}}$.  In other words, now we have proven that ${\displaystyle (R_{1}\cap R_{2}\rightarrow R_{1})\in F^{+}}$.

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References