Editing Semiring (section)

== Terminology ==
Some authors define semirings without the requirement for there to be a <math>0</math> or <math>1</math>. This makes the analogy between '''ring''' and {{em|semiring}} on the one hand and {{em|[[Group (mathematics)|group]]}} and {{em|[[semigroup]]}} on the other hand work more smoothly. These authors often use '''rig''' for the concept defined here.{{sfnp|Głazek|2002|p=7|ps=}}{{efn|[http://www.proofwiki.org/wiki/Definition:Rig For an example see the definition of rig on Proofwiki.org]}} This originated as a joke, suggesting that rigs are ri''n''gs without ''n''egative elements. (Akin to using ''[[rng (algebra)|rng]]'' to mean a r''i''ng without a multiplicative ''i''dentity.)

The term '''dioid''' (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzmann in 1972 to denote a semiring.{{refn|{{cite book|last=Kuntzmann|first=J.|title=Théorie des réseaux (graphes)|language=fr|zbl=0239.05101|location=Paris|publisher=Dunod|year=1972 }}}} (It is alternatively sometimes used for [[naturally ordered semiring]]s{{refn|[http://marcpouly.ch/pdf/internal_100712.pdf Semirings for breakfast], slide 17}} but the term was also used for idempotent subgroups by [[François Baccelli|Baccelli]] et al. in 1992.{{refn|{{cite book|last1=Baccelli|first1=François Louis|last2=Olsder|first2=Geert Jan|last3=Quadrat|first3=Jean-Pierre|last4=Cohen|first4=Guy|title=Synchronization and linearity. An algebra for discrete event systems|zbl=0824.93003|series=Wiley Series on Probability and Mathematical Statistics|location=Chichester|publisher=Wiley|year=1992 }}}})