Composition of relations

In the mathematics of binary relations, the composition of relations is the forming of a new binary relation R ; S from two given binary relations R and S. In the calculus of relations, the composition of relations is called relative multiplication,[1] and its result is called a relative product.[2]: 40  Function composition is the special case of composition of relations where all relations involved are functions.

The words uncle and aunt indicate a compound relation: for a person to be an uncle, he must be a brother of a parent (or a sister for an aunt). In algebraic logic it is said that the relation of Uncle (xUz) is the composition of relations "is a brother of" (xBy) and "is a parent of" (yPz).

Beginning with Augustus De Morgan,[3] the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition.[4]


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  ).[5]: 13 

Notational variationsEdit

The semicolon as an infix notation for composition of relations dates back to Ernst Schroder's textbook of 1895.[6] Gunther Schmidt has renewed the use of the semicolon, particularly in Relational Mathematics (2011).[2]: 40 [7] The use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in category theory,[8] as well as the notation for dynamic conjunction within linguistic dynamic semantics.[9]

A small circle   has been used for the infix notation of composition of relations by John M. Howie in his books considering semigroups of relations.[10] However, the small circle is widely used to represent composition of functions   which reverses the text sequence from the operation sequence. The small circle was used in the introductory pages of Graphs and Relations[5]: 18  until it was dropped in favor of juxtaposition (no infix notation). Juxtaposition   is commonly used in algebra to signify multiplication, so too, it can signify relative multiplication.

Further with the circle notation, subscripts may be used. Some authors[11] 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 ⨾ (U+2A3E FAT OPEN SEMICOLON) denotes left composition.[12][13]

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.


  • Composition of relations is associative:  
  • The converse relation of R ; S is (R ; S)T = ST ; RT. This property makes the set of all binary relations on a set a semigroup with involution.
  • The composition of (partial) functions (i.e. functional relations) is again a (partial) function.
  • If R and S are injective, then R ; S is injective, which conversely implies only the injectivity of R.
  • If R and S are surjective, then R ; S 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.

Composition in terms of matricesEdit

Finite binary relations are represented by logical matrices. The entries of these matrices are either zero or one, depending on whether the relation represented is false or true for the row and column corresponding to compared objects. Working with such matrices involves the Boolean arithmetic with 1 + 1 = 1 and 1 × 1 = 1. An entry in the matrix product of two logical matrices will be 1, then, only if the row and column multiplied have a corresponding 1. Thus the logical matrix of a composition of relations can be found by computing the matrix product of the matrices representing the factors of the composition. "Matrices constitute a method for computing the conclusions traditionally drawn by means of hypothetical syllogisms and sorites."[14]

Heterogeneous relationsEdit

Consider a heterogeneous relation RA × B; i.e. where A and B are distinct sets. Then using composition of relation R with its converse RT, there are homogeneous relations R RT (on A) and RT R (on B).

If ∀xAy ∈ B xRy (that is, R is a (left-)total relation), then ∀x xRRTx so that R RT is a reflexive relation or I ⊆ R RT where I is the identity relation {xIx : xA}. Similarly, if R is a surjective relation then

RT R ⊇ I = {xIx : xB}. In this case RR RT R. The opposite inclusion occurs for a difunctional relation.

The composition   is used to distinguish relations of Ferrer's type, which satisfy  


Let A = { France, Germany, Italy, Switzerland } and B = { French, German, Italian } with the relation R given by aRb when b is a national language of a. Since both A and B is finite, R can be represented by a logical matrix, assuming rows (top to bottom) and columns (left to right) are ordered alphabetically:


The converse relation RT corresponds to the transposed matrix, and the relation composition   corresponds to the matrix product   when summation is implemented by logical disjunction. It turns out that the 3×3 matrix   contains a 1 at every position, while the reversed matrix product computes as:


This matrix is symmetric, and represents a homogeneous relation on A.

Correspondingly,   is the universal relation on B, hence any two languages share a nation where they both are spoken (in fact: Switzerland). Vice versa, the question whether two given nations share a language can be answered using  .

Schröder rulesEdit

For a given set V, the collection of all binary relations on V forms a Boolean lattice ordered by inclusion (⊆). Recall that complementation reverses inclusion:   In the calculus of relations[15] it is common to represent the complement of a set by an overbar:  

If S is a binary relation, let   represent the converse relation, also called the transpose. Then the Schröder rules are


Verbally, one equivalence can be obtained from another: select the first or second factor and transpose it; then complement the other two relations and permute them.[5]: 15–19 

Though this transformation of an inclusion of a composition of relations was detailed by Ernst Schröder, in fact Augustus De Morgan first articulated the transformation as Theorem K in 1860.[4] He wrote


With Schröder rules and complementation one can solve for an unknown relation X in relation inclusions such as


For instance, by Schröder rule   and complementation gives   which is called the left residual of S by R .


Just as composition of relations is a type of multiplication resulting in a product, so some operations compare to division and produce quotients. Three quotients are exhibited here: left residual, right residual, and symmetric quotient. The left residual of two relations is defined presuming that they have the same domain (source), and the right residual presumes the same codomain (range, target). The symmetric quotient presumes two relations share a domain and a codomain.


  • Left residual:  
  • Right residual:  
  • Symmetric quotient:  

Using Schröder's rules, AXB is equivalent to XA B. Thus the left residual is the greatest relation satisfying AXB. Similarly, the inclusion YCD is equivalent to YD/C, and the right residual is the greatest relation satisfying YCD.[2]: 43–6 

One can practice the logic of residuals with Sudoku.[further explanation needed]

Join: another form of compositionEdit

A fork operator (<) has been introduced to fuse two relations c: HA and d: HB into c(<)d: HA × B. The construction depends on projections a: A × BA and b: A × BB, understood as relations, meaning that there are converse relations aT and bT. Then the fork of c and d is given by


Another form of composition of relations, which applies to general n-place relations for n ≥ 2, is 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. For example, in the query language SQL there is the operation Join (SQL).

See alsoEdit


  1. ^ Bjarni Jónssen (1984) "Maximal Algebras of Binary Relations", in Contributions to Group Theory, K.I. Appel editor American Mathematical Society ISBN 978-0-8218-5035-0
  2. ^ a b c Gunther Schmidt (2011) Relational Mathematics, Encyclopedia of Mathematics and its Applications, vol. 132, Cambridge University Press ISBN 978-0-521-76268-7
  3. ^ A. De Morgan (1860) "On the Syllogism: IV and on the Logic of Relations"
  4. ^ a b Daniel D. Merrill (1990) Augustus De Morgan and the Logic of Relations, page 121, Kluwer Academic ISBN 9789400920477
  5. ^ a b c Gunther Schmidt & Thomas Ströhlein (1993) Relations and Graphs, Springer books
  6. ^ Ernst Schroder (1895) Algebra und Logik der Relative
  7. ^ 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
  8. ^ Michael Barr & Charles Wells (1998) Category Theory for Computer Scientists Archived 2016-03-04 at the Wayback Machine, page 6, from McGill University
  9. ^ Rick Nouwen and others (2016) Dynamic Semantics §2.2, from Stanford Encyclopedia of Philosophy
  10. ^ John M. Howie (1995) Fundamentals of Semigroup Theory, page 16, LMS Monograph #12, Clarendon Press ISBN 0-19-851194-9
  11. ^ Kilp, Knauer & Mikhalev, p. 7
  12. ^ ISO/IEC 13568:2002(E), p. 23
  13. ^ Unicode character: Z Notation relational composition from
  14. ^ Irving Copilowish (December 1948) "Matrix development of the calculus of relations", Journal of Symbolic Logic 13(4): 193–203 Jstor link, quote from page 203
  15. ^ Vaughn Pratt The Origins of the Calculus of Relations, from Stanford University
  16. ^ De Morgan indicated contraries by lower case, conversion as M−1, and inclusion with )), so his notation was  
  17. ^ Gunther Schmidt and Michael Winter (2018): Relational Topology, page 26, Lecture Notes in Mathematics vol. 2208, Springer books, ISBN 978-3-319-74451-3


  • M. Kilp, U. Knauer, A.V. Mikhalev (2000) Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter,ISBN 3-11-015248-7.