Editing Monoid (section)

== Monoid structures ==

=== Submonoids ===
A '''submonoid''' of a monoid {{math|(''M'', •)}} is a [[subset]] {{math|''N''}} of {{math|''M''}} that is closed under the monoid operation and contains the identity element {{math|''e''}} of {{math|''M''}}.{{sfn|ps=|Jacobson|2009}}{{efn|Some authors omit the requirement that a submonoid must contain the identity element from its definition, requiring only that it have ''an'' identity element, which can be distinct from that of {{math|''M''}}.}} Symbolically, {{math|''N''}} is a submonoid of {{math|''M''}} if {{math|''e'' ∈ ''N'' ⊆ ''M''}}, and {{math|''x'' • ''y'' ∈ ''N''}} whenever {{math|''x'', ''y'' ∈ ''N''}}.  In this case, {{math|''N''}} is a monoid under the binary operation inherited from {{math|''M''}}.

On the other hand, if {{math|''N''}} is a subset of a monoid that is [[closure (mathematics)|closed]] under the monoid operation, and is a monoid for this inherited operation, then {{math|''N''}} is not always a submonoid, since the identity elements may differ. For example, the [[singleton set]] {{math|{{mset|0}}}} is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the [[nonnegative integer]]s.

=== Generators ===
A subset {{math|''S''}} of {{math|''M''}} is said to ''generate'' {{math|''M''}} if the smallest submonoid of {{math|''M''}} containing {{math|''S''}} is {{math|''M''}}. If there is a finite set that generates {{math|''M''}}, then {{math|''M''}} is said to be a '''finitely generated monoid'''.

=== 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''}}.

=== Partially commutative monoid ===
A monoid for which the operation is commutative for some, but not all elements is a [[trace monoid]]; trace monoids commonly occur in the theory of [[concurrent computation]].