Editing Semiring (section)

=== Complete semirings ===
A '''complete semiring''' is a semiring for which the additive monoid is a [[complete monoid]], meaning that it has an [[Finitary|infinitary]] sum operation <math>\Sigma_I</math> for any [[index set]] <math>I</math> and that the following (infinitary) distributive laws must hold:<ref name=Kuich11/>{{sfnp|Droste|Kuich|2009|pp=7–10|ps=}}<ref>{{cite book|last=Kuich|first=Werner|chapter=ω-continuous semirings, algebraic systems and pushdown automata|pages=[https://archive.org/details/automatalanguage0000ical/page/103 103–110]|title=Automata, Languages and Programming: 17th International Colloquium, Warwick University, England, July 16–20, 1990, Proceedings|volume=443|series=Lecture Notes in Computer Science|editor1-first=Michael S.|editor1-last=Paterson|publisher=[[Springer-Verlag]]|year=1990|isbn=3-540-52826-1|chapter-url=https://archive.org/details/automatalanguage0000ical/page/103 }}</ref>
: <math>{\textstyle\sum}_{i \in I}{\left(a \cdot a_i\right)} = a \cdot \left({\textstyle\sum}_{i \in I}{a_i}\right), \qquad {\textstyle\sum}_{i \in I}{\left(a_i \cdot a\right)} = \left({\textstyle\sum}_{i \in I}{a_i}\right) \cdot a.</math>
Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.{{sfnp|Sakarovitch|2009|p=471|ps=}}
For commutative, additively idempotent and simple semirings, this property is related to [[residuated lattice]]s.

==== Continuous semirings ====
A '''continuous semiring''' is similarly defined as one for which the addition monoid is a [[continuous monoid]]. That is, partially ordered with the [[Least-upper-bound property#Generalization to ordered sets|least upper bound property]], and for which addition and multiplication respect order and suprema.  The semiring <math>\N \cup \{ \infty \}</math> with usual addition, multiplication and order extended is a continuous semiring.{{refn|{{cite book|last1=Ésik|first1=Zoltán|last2=Leiß|first2=Hans|chapter=Greibach normal form in algebraically complete semirings|zbl=1020.68056|editor1-last=Bradfield|editor1-first=Julian|title=Computer science logic. 16th international workshop, CSL 2002, 11th annual conference of the EACSL, Edinburgh, Scotland, September 22–25, 2002. Proceedings|location=Berlin|publisher=[[Springer-Verlag]]|series=Lecture Notes in Computer Science|volume=2471|pages=135–150|year=2002 }}}}

Any continuous semiring is complete:<ref name=Kuich11/> this may be taken as part of the definition.{{sfnp|Sakarovitch|2009|p=471|ps=}}