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Binary operation
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==Terminology== More precisely, a binary operation on a [[Set (mathematics)|set]] <math>S</math> is a [[Map (mathematics)|mapping]] of the elements of the [[Cartesian product]] <math>S \times S</math> to <math>S</math>:<ref>{{harvnb|Rotman|1973|loc=pg. 1}}</ref><ref>{{harvnb|Hardy|Walker|2002|loc=pg. 176, Definition 67}}</ref><ref>{{harvnb|Fraleigh|1976|loc= pg. 10}}</ref> :<math>\,f \colon S \times S \rightarrow S.</math> If <math>f</math> is not a [[Function (mathematics)|function]] but a [[partial function]], then <math>f</math> is called a '''partial binary operation'''. For instance, division is a partial binary operation on the set of all [[real numbers]], because one cannot [[Division by zero|divide by zero]]: <math>\frac{a}{0}</math> is undefined for every real number <math>a</math>. In both [[model theory]] and classical [[universal algebra]], binary operations are required to be defined on all elements of <math>S \times S</math>. However, [[partial algebra]]s<ref name="Gratzer2008">{{cite book|author=George A. Grätzer|title=Universal Algebra|url=https://archive.org/details/isbn_9780387774862|url-access=registration|year=2008|publisher=Springer Science & Business Media|isbn=978-0-387-77487-9|at=Chapter 2. Partial algebras|edition=2nd}}</ref> generalize [[universal algebra]]s to allow partial operations. Sometimes, especially in [[computer science]], the term binary operation is used for any [[binary function]].
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