Editing Semiring (section)

== Generalizations ==
A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a [[semigroup]] rather than a monoid. Such structures are called {{em|hemirings}}{{sfnp|Golan|1999|p=1|loc=Ch 1|ps=}} or {{em|pre-semirings}}.{{sfnp|Gondran|Minoux|2008|p=22|loc=Ch 1, §4.2}} A further generalization are {{em|left-pre-semirings}},{{sfnp|Gondran|Minoux|2008|p=20|loc=Ch 1, §4.1}} which additionally do not require right-distributivity (or {{em|right-pre-semirings}}, which do not require left-distributivity).

Yet a further generalization are {{em|[[near-semiring]]s}}: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do [[ordinal number]]s form a [[near-semiring]], when the standard [[Ordinal arithmetic|ordinal addition and multiplication]] are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called [[Ordinal arithmetic#Natural operations|natural (or Hessenberg) operations]] instead.

In [[category theory]], a {{em|2-rig}} is a category with [[functor]]ial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the [[category of sets]] (or more generally, any [[topos]]) is a 2-rig.