Editing Kleene algebra (section)

== Generalization (or relation to other structures) ==
Kleene algebras are a particular case of [[closed semiring]]s, also called [[quasi-regular semiring]]s or [[Lehmann semiring]]s, which are semirings in which every element has at least one quasi-inverse satisfying the equation: ''a''* = ''aa''* + 1 = ''a''*''a'' + 1. This quasi-inverse is not necessarily unique.<ref name="Golan2003">{{cite book|author=Jonathan S. Golan|title=Semirings and Affine Equations over Them|url=https://books.google.com/books?id=jw4Hmgz5ETQC&pg=PA157|date=30 June 2003|publisher=Springer Science & Business Media|isbn=978-1-4020-1358-4|pages=157–159}}</ref><ref name="PoulyKohlas2012b"/> In a Kleene algebra, ''a''* is the least solution to the [[fixpoint]] equations: ''X'' = ''aX'' + 1 and ''X'' = ''Xa'' + 1.<ref name="PoulyKohlas2012b"/>

Closed semirings and Kleene algebras appear in [[algebraic path problem]]s, a generalization of the [[shortest path]] problem.<ref name="PoulyKohlas2012b">{{cite book|author1=Marc Pouly|author2=Jürg Kohlas|title=Generic Inference: A Unifying Theory for Automated Reasoning|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-01086-0|pages=232 and 248}}</ref>