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

Jeremy Siek, Indiana University (jsiek@osl.iu.edu)
Lie-Quan Lee, Indiana University (llee@cs.indiana.edu)
Andrew Lumsdaine, Indiana University (lums@osl.iu.edu)