Editing Monoid (section)

== Properties ==
The monoid axioms imply that the identity element {{math|''e''}} is unique: If {{math|''e''}} and {{math|''f''}} are identity elements of a monoid, then {{math|1=''e'' = ''ef'' = ''f''}}.

=== Products and powers ===
For each nonnegative integer {{math|''n''}}, one can define the product <math>p_n = \textstyle \prod_{i=1}^n a_i</math> of any sequence {{math|(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} of {{math|''n''}} elements of a monoid recursively: let {{math|1=''p''<sub>0</sub> = ''e''}} and let {{math|1=''p''<sub>''m''</sub> = ''p''<sub>''m''−1</sub> • ''a''<sub>''m''</sub>}} for {{math|1 ≤ ''m'' ≤ ''n''}}.

As a special case, one can define nonnegative integer powers of an element {{math|''x''}} of a monoid: {{math|1=''x''<sup>0</sup> = 1}} and {{math|1=''x''<sup>''n''</sup> = ''x''<sup>''n''−1</sup> • ''x''}} for {{math|''n'' ≥ 1}}.  Then {{math|1=''x''<sup>''m''+''n''</sup> = ''x''<sup>''m''</sup> • ''x''<sup>''n''</sup>}} for all {{math|''m'', ''n'' ≥ 0}}.

=== Invertible elements ===
An element {{math|''x''}} is called [[inverse element|invertible]] if there exists an element {{math|''y''}} such that {{math|1=''x'' • ''y'' = ''e''}} and {{math|1=''y'' • ''x'' = ''e''}}. The element {{math|''y''}} is called the inverse of {{math|''x''}}. Inverses, if they exist, are unique: if {{math|''y''}} and {{math|''z''}} are inverses of {{math|''x''}}, then by associativity {{math|1=''y'' = ''ey'' = (''zx'')''y'' = ''z''(''xy'') = ''ze'' = ''z''}}.{{sfn|ps=|Jacobson|2009|p=31|loc=§1.2}}

If {{math|''x''}} is invertible, say with inverse {{math|''y''}}, then one can define negative powers of {{math|''x''}} by setting {{math|1=''x''<sup>−''n''</sup> = ''y''<sup>''n''</sup>}} for each {{math|''n'' ≥ 1}}; this makes the equation {{math|1=''x''<sup>''m''+''n''</sup> = ''x''<sup>''m''</sup> • ''x''<sup>''n''</sup>}} hold for all {{math|''m'', ''n'' ∈ '''Z'''}}.

The set of all invertible elements in a monoid, together with the operation •, forms a [[group (mathematics)|group]].

=== Grothendieck group ===
{{main|Grothendieck group}}
Not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements {{math|''a''}} and {{math|''b''}} exist such that {{math|1=''a'' • ''b'' = ''a''}} holds even though {{math|''b''}} is not the identity element. Such a monoid cannot be embedded in a group, because in the group multiplying both sides with the inverse of {{math|''a''}} would get that {{math|1=''b'' = ''e''}}, which is not true. 

A monoid {{math|(''M'', •)}} has the [[cancellation property]] (or is cancellative) if for all {{math|''a''}}, {{math|''b''}} and {{math|''c''}} in {{math|''M''}}, the equality {{math|1=''a'' • ''b'' = ''a'' • ''c''}} implies {{math|1=''b'' = ''c''}}, and the equality {{math|1=''b'' • ''a'' = ''c'' • ''a''}} implies {{math|1=''b'' = ''c''}}. 

A commutative monoid with the cancellation property can always be embedded in a group via the ''Grothendieck group construction''. That is how the additive group of the integers (a group with operation {{math|+}}) is constructed from the additive monoid of natural numbers (a commutative monoid with operation {{math|+}} and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.

If a monoid has the cancellation property and is ''finite'', then it is in fact a group.{{efn|Proof: Fix an element {{math|''x''}} in the monoid. Since the monoid is finite, {{math|1=''x''<sup>''n''</sup> = ''x''<sup>''m''</sup>}} for some {{math|''m'' > ''n'' > 0}}. But then, by cancellation we have that {{math|1=''x''<sup>''m''−''n''</sup> = ''e''}} where {{math|''e''}} is the identity. Therefore, {{math|1=''x'' • ''x''<sup>''m''−''n''−1</sup> = ''e''}}, so {{math|''x''}} has an inverse.}}

The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. are closed under the operation and obviously include the identity). This means that the cancellative elements of any commutative monoid can be extended to a group.

The cancellative property in a monoid is not necessary to perform the Grothendieck construction – commutativity is sufficient. However, if a commutative monoid does not have the cancellation property, the homomorphism of the monoid into its Grothendieck group is not injective. More precisely, if {{math|1=''a'' • ''b'' = ''a'' • ''c''}}, then {{math|''b''}} and {{math|''c''}} have the same image in the Grothendieck group, even if {{math|''b'' ≠ ''c''}}. In particular, if the monoid has an [[absorbing element]], then its Grothendieck group is the [[trivial group]].

=== Types of monoids ===
An '''inverse monoid''' is a monoid where for every {{math|''a''}} in {{math|''M''}}, there exists a unique {{math|''a''<sup>−1</sup>}} in {{math|''M''}} such that {{math|1=''a'' = ''a'' • ''a''<sup>−1</sup> • ''a''}} and {{math|1=''a''<sup>−1</sup> = ''a''<sup>−1</sup> • ''a'' • ''a''<sup>−1</sup>}}. If an inverse monoid is cancellative, then it is a group.

In the opposite direction, a ''[[zerosumfree monoid]]'' is an additively written monoid in which {{math|1=''a'' + ''b'' = 0}} implies that {{math|1=''a'' = 0}} and {{math|1=''b'' = 0}}:{{sfn|ps=|Wehrung|1996}} equivalently, that no element other than zero has an additive inverse.