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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 $\struct {S, \circ}$ which satisfies the following properties:

\((\text S 0)\)   $:$   Closure      \(\ds \forall a, b \in S:\) \(\ds a \circ b \in S \)      
\((\text S 1)\)   $:$   Associativity      \(\ds \forall a, b, c \in S:\) \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)      
\((\text S 2)\)   $:$   Identity      \(\ds \exists e_S \in S: \forall a \in S:\) \(\ds e_S \circ a = a = a \circ e_S \)      

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.

Some sources present the monoid as $\struct {S, \circ, e_S}$ in order to place emphasis on the identity element, but this approach is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ as it complicates notation to little advantage.


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.