Editing Monoid (section)

== Definition ==

A [[set (mathematics)|set]] {{math|''S''}} equipped with a [[binary operation]] {{math|''S'' × ''S'' → ''S''}}, which we will denote {{math|•}}, is a '''monoid''' if it satisfies the following two axioms:

; Associativity: For all {{math|''a''}}, {{math|''b''}} and {{math|''c''}} in {{math|''S''}}, the equation {{math|1=(''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'')}} holds.
; Identity element: There exists an element {{math|''e''}} in {{math|''S''}} such that for every element {{math|''a''}} in {{math|''S''}}, the equalities {{math|1=''e'' • ''a'' = ''a''}} and {{math|1=''a'' • ''e'' = ''a''}} hold.

In other words, a monoid is a [[semigroup]] with an [[identity element]]. It can also be thought of as a [[magma (algebra)|magma]] with associativity and identity. The identity element of a monoid is unique.{{efn|If both {{math|''e''<sub>1</sub>}} and {{math|''e''<sub>2</sub>}} satisfy the above equations, then {{math|1=''e''<sub>1</sub> = ''e''<sub>1</sub> • ''e''<sub>2</sub> = ''e''<sub>2</sub>}}.}}  For this reason the identity is regarded as a [[Constant (mathematics)|constant]], i. e. {{math|0}}-ary (or nullary) operation. The monoid therefore is characterized by specification of the [[Tuple|triple]] {{math|(''S'', • , ''e'')}}.

Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by [[juxtaposition]]; for example, the monoid axioms may be written {{math|1=(''ab'')''c'' = ''a''(''bc'')}} and {{math|1=''ea'' = ''ae'' = ''a''}}. This notation does not imply that it is numbers being multiplied.

A monoid in which each element has an [[inverse element|inverse]] is a [[group (mathematics)|group]].