# Definition:Monoid

## Definition

A monoid is a semigroup with an identity element.

### Monoid Axioms

The properties that define a monoid can be gathered together as follows:

A monoid is an algebraic structure $\struct {S, \circ, e_S}$ which satisfies the following properties:

 $(S0)$ $:$ Closure $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b \in S$ $(S1)$ $:$ Associativity $\displaystyle \forall a, b, c \in S:$ $\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c$ $(S2)$ $:$ Identity $\displaystyle \exists e_S \in S: \forall a \in S:$ $\displaystyle e_S \circ a = a = a \circ e_S$

The element $e_S$ is called the identity element.

## Also known as

Some treatments of group theory and abstract algebra do not introduce the term monoid, but simply discuss semigroups which happen to have an identity element.

## Examples

### Operation Defined as $x + y + x y$ on Real Numbers

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

$\forall x, y \in \R: x \circ y := x + y + x y$

Then $\struct {\R, \circ}$ is a monoid whose identity is $0$.

## Also see

• Results about monoids can be found here.