Composition of relations
If and are two binary relations, then their composition is the relation
In other words, is defined by the rule that says if and only if there is an element such that (i.e. and ).
In particular fields, authors might denote by R ∘ S what is defined here to be S ∘ R. The convention chosen here is such that function composition (with the usual notation) is obtained as a special case, when R and S are functional relations. Some authors prefer to write and explicitly when necessary, depending whether the left or the right relation is the first one applied.
A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition. This use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category theory, as well as the notation for dynamic conjunction within linguistic dynamic semantics. The semicolon notation (with this semantic) was introduced by Ernst Schröder in 1895.
The binary relations are sometimes regarded as the morphisms in a category Rel which has the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. The category Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of this is found in the theory of allegories.
Composition of relations is associative.
The composition of (partial) functions (i.e. functional relations) is again a (partial) function.
If R and S are injective, then S ∘ R is injective, which conversely implies only the injectivity of R.
If R and S are surjective, then S ∘ R is surjective, which conversely implies only the surjectivity of S.
The set of binary relations on a set X (i.e. relations from X to X) together with (left or right) relation composition forms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element.
Join: another form of composition
Other forms of composition of relations, which apply to general n-place relations instead of binary relations, are found in the join operation of relational algebra. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.
Composition in terms of matrices
If is a relation between two finite sets then one obtains an associated adjacency matrix (see here). Then if and are two relations which can be composed, the matrix is exactly the matrix product , where it is understood that . (Note also the reversal of order of and .)
- Kilp, Knauer & Mikhalev, p. 7
- ISO/IEC 13568:2002(E), p. 23
- http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf, p. 6
- Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 24. ISBN 978-0-521-63107-5. A free HTML version of the book is available at http://www.cs.man.ac.uk/~pt/Practical_Foundations/
- M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.