Editing Monoid (section)

=== Commutative monoid ===
A monoid whose operation is [[commutative]] is called a '''commutative monoid''' (or, less commonly, an '''abelian monoid'''). Commutative monoids are often written additively. Any commutative monoid is endowed with its ''algebraic'' [[preorder]]ing {{math|≤}}, defined by {{math|''x'' ≤ ''y''}} if there exists {{math|''z''}} such that {{math|1=''x'' + ''z'' = ''y''}}.{{sfn|ps=|Gondran|Minoux|2008|p=13}} An ''order-unit'' of a commutative monoid {{math|''M''}} is an element {{math|''u''}} of {{math|''M''}} such that for any element {{math|''x''}} of {{math|''M''}}, there exists {{math|''v''}} in the set generated by {{math|''u''}} such that {{math|''x'' ≤ ''v''}}. This is often used in case {{math|''M''}} is the [[Ordered group|positive cone]] of a [[Partially ordered set|partially ordered]] [[abelian group]] {{math|''G''}}, in which case we say that {{math|''u''}} is an order-unit of {{math|''G''}}.