A relation r {\displaystyle r} on schema R {\displaystyle R} decomposes losslessly onto schemas R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} if π R 1 ( r ) ⋈ π R 2 ( r ) = r {\displaystyle \pi _{R_{1}}(r)\bowtie \pi _{R_{2}}(r)=r} , that is r {\displaystyle r} is the natural join of its projections onto the smaller schemas. A pair ( R 1 , R 2 ) {\displaystyle (R_{1},R_{2})} is a lossless-join decomposition of R {\displaystyle R} or said to have a lossless join with respect to a set of functional dependencies F {\displaystyle F} if any relation r ( R ) {\displaystyle r(R)} that satisfies F {\displaystyle F} decomposes losslessly onto R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} .3
Decompositions into more than two schemas can be defined in the same way.4
A decomposition R = R 1 ∪ R 2 {\displaystyle R=R_{1}\cup R_{2}} has a lossless join with respect to F {\displaystyle F} if and only if the closure of R 1 ∩ R 2 {\displaystyle R_{1}\cap R_{2}} includes R 1 ∖ R 2 {\displaystyle R_{1}\setminus R_{2}} or R 2 ∖ R 1 {\displaystyle R_{2}\setminus R_{1}} . In other words, one of the following must hold:5
Multiple sub-schemas R 1 , R 2 , . . . , R n {\displaystyle R_{1},R_{2},...,R_{n}} have a lossless join if there is some way in which we can repeatedly perform lossless joins until all the schemas have been joined into a single schema. Once we have a new sub-schema made from a lossless join, we are not allowed to use any of its isolated sub-schema to join with any of the other schemas. For example, if we can do a lossless join on a pair of schemas R i , R j {\displaystyle R_{i},R_{j}} to form a new schema R i , j {\displaystyle R_{i,j}} , we use this new schema (rather than R i {\displaystyle R_{i}} or R j {\displaystyle R_{j}} ) to form a lossless join with another schema R k {\displaystyle R_{k}} (which may already be joined (e.g., R k , l {\displaystyle R_{k,l}} )).
Pohler, K (2015). "Lossless-Join Decomposition: applications in quantitative computing metrics". International Journal of Applied Computer Science. 21 (4): 190–212. ↩
Elmasri, Ramez (2016). Fundamentals of database systems (Seventh ed.). Hoboken, NJ: Pearson. p. 461. ISBN 978-0133970777. 978-0133970777 ↩
Maier, David (1983). The theory of relational databases (PDF). Computer Science Press. p. 101. ISBN 0-914894-42-0. Retrieved 16 August 2024. 0-914894-42-0 ↩
Ullman, Jeffrey D. (1988). Principles of Database and Knowledge-base Systems (PDF) (1 ed.). Computer Science Press. p. 397. ISBN 0-88175188-X. Retrieved 16 August 2024. 0-88175188-X ↩
"Lossless-Join Decomposition". Cs.sfu.ca. Retrieved 2016-02-07. http://www.cs.sfu.ca/CourseCentral/354/zaiane/material/notes/Chapter7/node7.html ↩
"www.data-e-education.com - Lossless Join Decomposition". Archived from the original on 2014-02-21. Retrieved 2014-02-12. https://web.archive.org/web/20140221081148/http://www.data-e-education.com/E121_Lossless_Join_Decomposition.html ↩