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Monoid

A Monoid is a concept that describes a simple kind of algebraic system. A monoid consists of a set of elements S, a binary operation, and an identity element. The C++ representation of a monoid consists of a function object that implements the binary operation, a set of objects that represent the elements of S, and an object that represents the identity element.

Refinement of

The element type must be a model of Assignable and CopyConstructible. The function object type must be a model of BinaryFunction.

Notation

X Is the element type of the Monoid.

a, b

are objects of type X.

op

Is the function object implementing the Monoid operation.

i

is an object of type X and is the identity element for the Monoid.

Valid Expressions

op(a, b)

See below for semantics.
Return type: X

a == b

Returns true if a and b represent the same element of S.
Return type: bool

a != b

Returns true if a and b represent different elements of S.
Return type: bool

Invariants

  • Closure
    The result of op(a, b) is also an element of S.

  • Associativity
    op(op(a, b), c) == op(a, op(b, c))

  • Definition of Identity Element
    op(a, i) == a

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