Editing Monoid (section)

== Complete monoids ==
A '''complete monoid''' is a commutative monoid equipped with an [[Finitary|infinitary]] sum operation <math>\Sigma_I</math> for any [[index set]] {{math|''I''}} such that{{sfn|ps=|Droste|Kuich|2009|pp=7–10}}{{sfn|ps=|Hebisch|1992}}{{sfn|ps=|Kuich|1990}}{{sfn|Kuich|2011}}
: <math>\sum_{i \in \emptyset}{m_i} =0;\quad \sum_{i \in \{j\}}{m_i} = m_j;\quad \sum_{i \in \{j, k\}}{m_i} = m_j+m_k \quad \text{ for } j\neq k</math>
and
: <math>\sum_{j \in J}{\sum_{i \in I_j}{m_i}} = \sum_{i \in I} m_i \quad  \text{ if } \bigcup_{j\in J} I_j=I \text{ and } I_j \cap I_{j'} = \emptyset \quad \text{ for } j\neq j'</math>.

An '''ordered commutative monoid''' is a commutative monoid {{math|''M''}} together with a [[partial ordering]] {{math|≤}} such that {{math|''a'' ≥ 0}} for every {{math|''a'' ∈ ''M''}}, and {{math|''a'' ≤ ''b''}} implies {{math|''a'' + ''c'' ≤ ''b'' + ''c''}} for all {{math|''a'', ''b'', ''c'' ∈ ''M''}}.

A '''continuous monoid''' is an ordered commutative monoid {{math|(''M'', ≤)}} in which every [[directed set|directed subset]] has a [[least upper bound]], and these least upper bounds are compatible with the monoid operation:
: <math>a + \sup S = \sup(a + S)</math>
for every {{math|''a'' ∈ ''M''}} and directed subset {{math|''S''}} of {{math|''M''}}.

If {{math|(''M'', ≤)}} is a continuous monoid, then for any index set {{math|''I''}} and collection of elements {{math|(''a''{{sub|''i''}}){{sub|''i''∈''I''}}}}, one can define  
: <math> \sum_I a_i = \sup_{\text{finite } E \subset I} \; \sum_E a_i, </math>
and {{math|''M''}} together with this infinitary sum operation is a complete monoid.{{sfn|Kuich|2011}}