Editing Relational algebra (section)

== Implementations ==

The first query language to be based on Codd's algebra was Alpha, developed by Dr. Codd himself. Subsequently, [[ISBL]] was created, and this pioneering work has been acclaimed by many authorities<ref>{{cite web |title = Edgar F. Codd - A.M. Turing Award Laureate |url = https://amturing.acm.org/award_winners/codd_1000892.cfm |author = C. J. Date |website = amturing.acm.org |access-date = 2020-12-27 }}</ref> as having shown the way to make Codd's idea into a useful language. [[Business System 12]] was a short-lived industry-strength relational DBMS that followed the ISBL example.

In 1998 [[Christopher J. Date|Chris Date]] and [[Hugh Darwen]] proposed a language called '''Tutorial D''' intended for use in teaching relational database theory, and its query language also draws on ISBL's ideas.<ref>{{cite web |title = Databases, Types, and the Relational model: The Third Manifesto |author = C. J. Date and Hugh Darwen |url = https://www.dcs.warwick.ac.uk/~hugh/TTM/DTATRM.pdf |access-date = 2024-07-04}}</ref> Rel is an implementation of '''Tutorial D'''. Bmg is an implementation of relational algebra in Ruby which closely follows the principles of '''Tutorial D''' and ''The Third Manifesto''.<ref>{{cite web |title = Bmg documentation |url = https://www.relational-algebra.dev/ |access-date = 2024-07-04}}</ref>

Even the query language of [[SQL]] is loosely based on a relational algebra, though the operands in SQL ([[table (database)|table]]s) are not exactly [[relation (database)|relation]]s and several useful theorems about the relational algebra do not hold in the SQL counterpart (arguably to the detriment of optimisers and/or users). The SQL table model is a bag ([[multiset]]), rather than a set. For example, the expression <math>(R \cup S) \setminus T = (R \setminus T) \cup (S \setminus T)</math> is a theorem for relational algebra on sets, but not for relational algebra on bags.<ref name="Garcia-MolinaUllman2009">{{cite book |author1 = [[Hector Garcia-Molina]] |author2 = [[Jeffrey Ullman|Jeffrey D. Ullman]] |author3 = [[Jennifer Widom]] |title = Database systems: the complete book |year = 2009 |publisher = Pearson Prentice Hall |isbn = 978-0-13-187325-4 |edition = 2nd }}</ref>