Composition of relations

If ${\displaystyle R\subseteq X\times Y}$ and ${\displaystyle S\subseteq Y\times Z}$ are two binary relations, then their composition ${\displaystyle R;S}$ is the relation

${\displaystyle R;S=\{(x,z)\in X\times Z\mid \exists y\in Y:(x,y)\in R\land (y,z)\in S\}.}$

In other words, ${\displaystyle R;S\subseteq X\times Z}$ is defined by the rule that says ${\displaystyle (x,z)\in R;S}$ if and only if there is an element ${\displaystyle y\in Y}$ such that ${\displaystyle x\,R\,y\,S\,z}$ (i.e. ${\displaystyle (x,y)\in R}$ and ${\displaystyle (y,z)\in S}$).[5]^:13

• Composition of relations is associative: ${\displaystyle R;(S;T)\ =\ (R;S);T.}$
• The converse relation of R ; S is (R ; S)^T = S^T ; R^T. 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.

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]

Consider a heterogeneous relation R ⊆ A × B. Then using composition of relation R with its converse R^T, there are homogeneous relations R R^T (on A) and R^T R (on B).

If ∀x ∈ A ∃y ∈ B xRy (R is a total relation), then ∀x xRR^Tx so that R R^T is a reflexive relation or I ⊆ R R^T where I is the identity relation {xIx : x ∈ A}. Similarly, if R is a surjective relation then

R^T R ⊇ I = {xIx : x ∈ B}. In this case R ⊆ R R^T R. The opposite inclusion occurs for a difunctional relation.

The composition ${\displaystyle {\bar {R}}^{T}R}$ is used to distinguish relations of Ferrer's type, which satisfy ${\displaystyle R{\bar {R}}^{T}R=R.}$

Example

Composition of R with R^T produces the relation with this graph, Switzerland connected to the other countries (loops not shown).

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. The logical matrix for R is given by

${\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\1&1&1\end{pmatrix}}.}$ Using the converse relation R^T, two questions can be answered: "Is there a translator?" has answer ${\displaystyle R^{T}R=B\times B,}$ the universal relation on B. The international question, "Does he speak my language?" is answered by ${\displaystyle RR^{T}={\begin{pmatrix}1&0&0&1\\0&1&0&1\\0&0&1&1\\1&1&1&1\end{pmatrix}}.}$ This symmetric matrix, representing a homogeneous relation on A, is associated with the star graph S₃ augmented by a loop at each node.

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: ${\displaystyle A\subset B\implies B^{\complement }\subseteq A^{\complement }.}$ In the calculus of relations[15] it is common to represent the complement of a set by an overbar: ${\displaystyle {\bar {A}}=A^{\complement }.}$

If S is a binary relation, let ${\displaystyle S^{T}}$ represent the converse relation, also called the transpose. Then the Schröder rules are

${\displaystyle QR\subseteq S\quad \equiv \quad Q^{T}{\bar {S}}\subseteq {\bar {R}}\quad \equiv \quad {\bar {S}}R^{T}\subseteq {\bar {Q}}.}$

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

${\displaystyle LM\subseteq N\implies {\bar {N}}M^{T}\subseteq {\bar {L}}.}$[16]

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

${\displaystyle RX\subseteq S\quad {\text{and}}\quad XR\subseteq S.}$

For instance, by Schröder rule ${\displaystyle RX\subseteq S\implies R^{T}{\bar {S}}\subseteq {\bar {X}},}$ and complementation gives ${\displaystyle X\subseteq {\overline {R^{T}{\bar {S}}}},}$ 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 compositions 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.

Definitions:

• Left residual: ${\displaystyle A\backslash B\ =\ {\overline {A^{T}{\bar {B}}}}}$
• Right residual: ${\displaystyle D/C\ =\ {\overline {{\bar {D}}C^{T}}}}$
• Symmetric quotient: ${\displaystyle \operatorname {syq} (E,F)={\overline {E^{T}{\bar {F}}}}\cap {\overline {{\bar {E}}^{T}F}}}$

Using Schröder's rules, AX ⊆ B is equivalent to X ⊆ A${\displaystyle \backslash }$B. Thus the left residual is the greatest relation satisfying AX ⊆ B. Similarly, the inclusion YC ⊆ D is equivalent to Y ⊆ D/C, and the right residual is the greatest relation satisfying YC ⊆ D.[2]^:43-6

A fork operator (<) has been introduced to fuse two relations c: H → A and d: H → B into c(<)d: H → A × B. The construction depends projections a: A × B → A and b: A × B → B, understood as relations, meaning that there are converse relations a^T and b^T. Then the fork of c and d is given by

${\displaystyle c(<)d\$ :=\ c;a^{T}\cap \ d;b^{T}.} [17]

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).

• Demonic composition
• Friend of a friend

• 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.