Composition of relations
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In mathematics, the composition of binary relations is a concept of forming a new relation S o R from two given relations R and S, having as its most well-known special case the composition of functions.
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[edit] Definition
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 o S what is defined here to be S o 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[1] 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 semicolon with Unicode code point U+2A3E[2]) denotes left composition. This use of semicolon coincides with the notation for function composition used in Category theory. Because U+2A3E is hard to distinguish from a normal semicolon in some fonts at small sizes, its "capital" version ⨟ (U+2A1F[3]) may be preferable in some fonts. In non-Unicode LaTeX, the symbol may obtain using the \fatsemi macro from the stmaryrd package.
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.
[edit] Properties
Composition of relations is associative.
The inverse relation of S o R is (S o R)-1 = R-1 o S-1.
The compose of (partial) functions (i.e. functional relations) is again a (partial) function.
If R and S are injective, then S o R is injective, which conversely implies only the injectivity of R.
If R and S are surjective, then S o 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.
[edit] 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.
[edit] See also
[edit] Notes
- ^ Klip, Knauer & Mikhalev, p. 7
- ^ http://www.fileformat.info/info/unicode/char/2a3e/index.htm
- ^ http://www.fileformat.info/info/unicode/char/2a1f/index.htm
[edit] References
- 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 3110152487.
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