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