Editing Monoid (section)

=== 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.