Editing Relational algebra (section)

=== Transitive closure ===
Although relational algebra seems powerful enough for most practical purposes, there are some simple and natural operators on [[relation (database)|relation]]s that cannot be expressed by relational algebra. One of them is the [[transitive closure]] of a binary relation. Given a domain ''D'', let binary relation ''R'' be a subset of ''D''×''D''. The transitive closure ''R<sup>+</sup>'' of ''R'' is the smallest subset of ''D''×''D'' that contains ''R'' and satisfies the following condition:

:<math>\forall x \forall y \forall z \left( (x,y) \in R^+ \wedge (y,z) \in R^+ \Rightarrow (x,z) \in R^+ \right)</math>

It can be proved using the fact that there is no relational algebra expression ''E''(''R'') taking ''R'' as a variable argument that produces ''R''<sup>+</sup>.<ref>{{cite journal|title=Universality of data retrieval languages|journal=Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages|year=1979|first=Alfred V.|last=Aho|author2=Jeffrey D. Ullman |pages=110–119|doi=10.1145/567752.567763|s2cid=3242505 |doi-access=free}}</ref>

SQL however officially supports such [[Hierarchical and recursive queries in SQL|fixpoint queries]] since 1999, and it had vendor-specific extensions in this direction well before that.