Binary operation: Difference between revisions
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imported>Jochen Burghardt →Terminology: suggest to remove incomprehensible sentence |
imported>Rgdboer m →Terminology: lk George Grätzer |
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:<math>\,f \colon S \times S \rightarrow S.</math> | :<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. | 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|author-link=George 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]]. | Sometimes, especially in [[computer science]], the term binary operation is used for any [[binary function]]. | ||
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Binary operations are often written using [[infix notation]] such as <math>a \ast b</math>, <math>a+b</math>, <math>a \cdot b</math> or (by [[Juxtaposition#Mathematics|juxtaposition]] with no symbol) <math>ab</math> rather than by functional notation of the form <math>f(a, b)</math>. Powers are usually also written without operator, but with the second argument as [[superscript]]. | Binary operations are often written using [[infix notation]] such as <math>a \ast b</math>, <math>a+b</math>, <math>a \cdot b</math> or (by [[Juxtaposition#Mathematics|juxtaposition]] with no symbol) <math>ab</math> rather than by functional notation of the form <math>f(a, b)</math>. Powers are usually also written without operator, but with the second argument as [[superscript]]. | ||
Binary operations are sometimes written using prefix or | Binary operations are sometimes written using prefix or postfix notation, both of which dispense with parentheses. They are also called, respectively, [[Polish notation]] <math>\ast a b</math> and [[reverse Polish notation]] <math>a b \ast</math>. | ||
== Binary operations as ternary relations == | == Binary operations as ternary relations == | ||