Definition:Monoid

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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.


Sources