DescriptionA Monoid Operation is a special sort of Binary Function. A Binary Function must satisfy three conditions in order to be a Monoid Operation. First, its first argument type and second argument type must be the same, and its result type must be the same as its argument type. Second, there must be an identity element. Third, the operation must be associative. Examples of Monoid Operations are addition and multiplication. 
Refinement ofBinary Function
DefinitionsA type F that is a model of binary function is associative if F's first argument type, second argument type, and result type are the same, and if, for every object f of type F and for every objects x, y, and z of F's argument type, f(x, f(y, z)) is the same as f(f(x, y), z). 
Valid ExpressionsIn addition to the expressions described in the Binary Function requirements, the following expressions must be valid.
 A monoid is one of three closely related algebraic structures. A semigroup is a set S, and a binary operation *, with the properties that * is closed on S (that is, if x and y are elements of S then x * y is also a member of S) and that * is associative (that is, if x, y, and z are elements of S, then x * (y * z) = (x * y) * z). A monoid is a semigroup that has an identity element. That is, there exists some element id such that, for all x in S, x * id = id * x = x. Finally, a group is a monoid with the property that every element has an inverse. That is, for every x in S, there exists an element xi such that x * xi = xi * x = id. As an example, the set of real numbers under multiplication is a monoid (the identity element is 1), but it isn't a group. It isn't a group because 0 has no inverse.
 Mathematics textbooks typically write this as an equation, instead of using words like "is the same as". We can't use equality in this definition, however, because F's argument type might not be equality comparable. If F's argument type is equality comparable, however, then these two expression are expected to be equal: the condition of associativity becomes f(x, f(y, z)) == f(f(x, y), z)
 This is implemented as an overloaded function. The function identity_element is defined, in the standard header functional, and the nonstandard backward-compatibility header function.h, for arguments of type plus<T> and multiplies<T>. If you define a new Monoid Operation F (matrix multiplication, for example), you must overload identity_element for arguments of type F. The identity_element function is an SGI extension; it is not part of the C++ standard.
 Associativity is not the same as commutativity. That is, the requirement that x * (y * z) == (x * y) * z is completely unrelated to the requirement that x * y == y * x. Monoid operations are required to be associative, but they are not required to be commutative. As an example, square matrices under multiplication form a monoid even though matrix multiplication is not commutative.
See alsoBinary Function, plus, multiplies
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