Editing Binary relation (section)

=== Matrix representation ===
Binary relations over sets <math>X</math> and <math>Y</math> can be represented algebraically by [[Logical matrix|logical matrices]] indexed by <math>X</math> and <math>Y</math> with entries in the [[Boolean semiring]] (addition corresponds to OR and multiplication to AND) where [[matrix addition]] corresponds to union of relations, [[matrix multiplication]] corresponds to composition of relations (of a relation over <math>X</math> and <math>Y</math> and a relation over <math>Y</math> and <math>Z</math>),<ref>{{cite newsgroup |title=quantum mechanics over a commutative rig |author=John C. Baez |author-link=John C. Baez |date=6 Nov 2001 |newsgroup=sci.physics.research |message-id=9s87n0$iv5@gap.cco.caltech.edu |url=https://groups.google.com/d/msg/sci.physics.research/VJNPMCfreao/TMKt9tFYNwEJ |access-date=November 25, 2018}}</ref> the [[Hadamard product (matrices)|Hadamard product]] corresponds to intersection of relations, the [[zero matrix]] corresponds to the empty relation, and the [[matrix of ones]] corresponds to the universal relation. Homogeneous relations (when <math>X = Y</math>) form a [[matrix semiring]] (indeed, a [[matrix semialgebra]] over the Boolean semiring) where the [[identity matrix]] corresponds to the identity relation.<ref name="droste">Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. ''Handbook of Weighted Automata'', 3–28. {{doi|10.1007/978-3-642-01492-5_1}}, pp. 7-10</ref>