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English

binary relation (plural binary relations)

(set theory, order theory, "on" a set A) A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A, alternatively written as A2).
Synonym: homogeneous relation
- 1978, George Grätzer, General Lattice Theory, Academic Press, page 1:
  A partially ordered set $⟨ A, ϱ ⟩$ consists of a nonvoid set $A$ and a binary relation $ϱ$ on $A$ , such that $ϱ$ satisfies properties (P1)-(P3).
- 1999, James C. Moore, Mathematical Methods for Economic Theory 1, Springer, page 24:
  1.30. Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.
  It should be noted that a binary relation may be irreflexive and negatively transitive without being transitive; as an example, consider the standard inequality relation (≠).
- 2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1:
  Definition If E is a non-empty set then by an order on E we mean a binary relation on E that is reflexive, anti-symmetric, and transitive.
(set theory, order theory, "on" or "between" sets A and B) A subset of the Cartesian product $A \times B$ .
Synonym: heterogeneous relation

If $R$ is a relation between $A$ and $B$ , then $A$ is called the domain of the relation, and $B$ is called the codomain of the relation.
For a binary relation $R$ , the notation $a R b$ signifies that $(a, b) \in R$ , and one may say that $a$ is in binary relation $R$ to $b$ .

order theory