Binary relation

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

A two-place predicate on a given set. The term is sometimes used to denote a subset of the set $A\times A$ of ordered pairs $(a,b)$ of elements of a given set $A$. A binary relation is a special case of a relation. Let $R\subseteq A\times A$. If $(a,b)\in R$, then one says that the element $a$ is in binary relation $R$ to the element $b$. An alternative notation for $(a,b)\in R$ is $aRb$.

The empty subset $\emptyset$ in $A\times A$ and the set $A\times A$ itself are called, respectively, the nil relation and the universal relation in the set $A$. The diagonal of the set $A\times A$, i.e. the set $\Delta=\{(a,a)\colon a\in A\}$, is the equality relation or the identity binary relation in $A$.

Let $R,R_1,R_2$ be binary relations in a set $A$. In addition to the set-theoretic operations of union $R_1\cup R_2$, intersection $R_1\cap R_2$, and negation or complementation $R'=(A\times A)\setminus R$, one has the inversion (also converse or transpose)

$$R^{-1}=\{(a,b)\colon(b,a)\in R\},$$

as well as the operation of multiplication (or composition):

$$R_1\circ R_2=\{(a,c) \in A\times A\colon(\exists b\in A)(aR_1b\text{ and }bR_2c)\}.$$

The binary relation $R^{-1}$ is said to be the inverse of $R$. Multiplication of binary relations is associative, but as a rule not commutative.

A binary relation $R$ in $A$ is said to be 1) reflexive if $\Delta\subseteq R$; 2) transitive if $R\circ R\subseteq R$; 3) symmetric if $R^{-1}\subseteq R$; and 4) anti-symmetric if $R\cap R^{-1}\subseteq\Delta$. If a binary relation has some of the properties 1), 2), 3) or 4), the inverse relation $R^{-1}$ has these properties as well. The binary relation $R\subseteq A\times A$ is said to be functional if $R^{-1}\circ R\subseteq\Delta$.

The most important types of binary relations are equivalences, order relations (total and partial), and functional relations.

Comments

More generally, a binary relation $R$ may be defined between two sets $A$ and $B$ as a two-place predicate or realised as a subset of the Cartesian product $A \times B$. As before we may speak of the union, intersection or negation of $R$ as a relation on $A \times B$. The transpose $R^t$ is now a relation on $B \times A$. Given relations $R_1$ on $A\times B$ and $R_2$ on $B\times C$, the composition $R_1 \circ R_2$ is a relation on $A \times C$:

$$R_1\circ R_2=\{(a,c) \in A \times C\colon(\exists b\in B)(aR_1b\text{ and }bR_2c)\}.$$

A relation $R$ on $A\times B$ is functional if for each $a \in A$ there is at most one $b \in B$ such that $a R b$. Such a relation defines a (partial) function from $A$ to $B$: cf. Functional relation.

References

[a1]	P. R. Halmos, Naive Set Theory, Springer (1960) ISBN 0-387-90092-6

Comments

The set $\mathcal{R}(A)$ of binary relations on $A$ form a monoid under composition of relations with the equality relation as identity element and the empty relation as a zero; indeed, it is a Baer semigroup.

References

[b1]	T.S. Blyth Lattices and Ordered Algebraic Structures Springer (2006) ISBN 184628127X
[b2]	R. Fraïssé, Theory of Relations, Studies in Logic and the Foundations of Mathematics, Elsevier (2011) ISBN 0080960413