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More precisely, a binary operation on a Set ''S'' is a Binary Function from ''S'' and ''S'' to ''S'', in other words a function ''f'' from the Cartesian Product ''S'' × ''S'' to ''S''. Sometimes, especially in Computer Science , the term is used for any Binary Function . That ''f'' takes values in the same set ''S'' that provides its arguments is the property of Closure . Binary operations are the keystone of algebraic structures studied in , Monoid s, Semigroup s, Ring s, and more. Most generally, a '' Magma '' is a set together with any binary operation defined on it. Many binary operations of interest in both algebra and formal logic are Commutative or Associative . Many also have Identity Element s and Inverse Element s.
Examples of operations that are not Commutative are Subtraction (-), Division (/), Exponentiation (^), and Super-exponentiation (@).
Sometimes they are even written just by Juxtaposition : ''ab''. They can also be expressed using prefix or postfix notations. A prefix notation, Polish Notation , dispenses with parentheses; it is probably more often encountered now in its postfix form, Reverse Polish Notation . EXTERNAL BINARY OPERATIONS An external binary operation is a binary function from ''K'' and ''S'' to ''S''. This differs from a binary operation in the strict sense in that ''K'' need not be ''S''; its elements come from ''outside''. An example of an external binary operation is Scalar Multiplication in Linear Algebra . Here ''K'' is a Field and ''S'' is a Vector Space over that field. An external binary operation may alternatively be viewed as an Action ; ''K'' is acting on ''S''. |
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