Lossless-Join Decomposition

Definition

A relation ${\displaystyle r}$ on schema ${\displaystyle R}$decomposes losslessly onto schemas ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ if ${\displaystyle \pi _{R_{1}}(r)\bowtie \pi _{R_{2}}(r)=r}$, that is ${\displaystyle r}$ is the natural join of its projections onto the smaller schemas. A pair ${\displaystyle (R_{1},R_{2})}$ is a lossless-join decomposition of ${\displaystyle R}$ or said to have a lossless join with respect to a set of functional dependencies${\displaystyle F}$ if any relation ${\displaystyle r(R)}$ that satisfies ${\displaystyle F}$ decomposes losslessly onto ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$.