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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| Copyright © 2000-2001 | Jeremy Siek, Indiana University ([email protected])
Lie-Quan Lee, Indiana University ([email protected])
Andrew Lumsdaine, Indiana University ([email protected]) |