binary relation

plural binary relations

1. "on" a set AmathematicsA subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A, alternatively written as A²).

   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.

   2005, T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, page 1

   2 more examples

   A partially ordered set #92;langleA,#92;varrho#92;rangle consists of a nonvoid set A and a binary relation #92;varrho on A, such that #92;varrho satisfies properties (P1)-(P3).

   1978, George Grätzer, General Lattice Theory, Academic Press, page 1

   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 (≠).

   1999, James C. Moore, Mathematical Methods for Economic Theory 1, Springer, page 24

   Synonyms: homogeneous relation

2. "on" or "between" sets A and BmathematicsA subset of the Cartesian product A×B.

   Synonyms: heterogeneous relation