# Semiring

In abstract algebra, a **semiring** is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

The term **rig** is also used occasionally^{[1]}—this originated as a joke, suggesting that rigs are ri*n*gs without *n*egative elements, similar to using *rng* to mean a r*i*ng without a multiplicative *i*dentity.

Tropical semirings are an active area of research, linking algebraic varieties with piecewise linear structures.^{[2]}

Compared to a ring, a semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoid, not a commutative group. In a ring, the additive inverse requirement implies the existence of a multiplicative zero, so here it must be specified explicitly. If a semiring's multiplication is commutative, then it is called a **commutative semiring**.^{[6]}

There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between *ring* and *semiring* on the one hand and *group* and *semigroup* on the other hand work more smoothly. These authors often use *rig* for the concept defined here.^{[note 1]}

One can generalize the theory of (associative) algebras over commutative rings directly to a theory of algebras over commutative semirings.^{[citation needed]}

Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.^{[22]}

Any continuous semiring is complete:^{[19]} this may be taken as part of the definition.^{[22]}

A **star semiring** (sometimes spelled **starsemiring**) is a semiring with an additional unary operator ^{∗},^{[7]}^{[17]}^{[24]}^{[25]} satisfying

A **Kleene algebra** is a star semiring with idempotent addition and some additional axioms. They are important in the theory of formal languages and regular expressions.^{[17]}

In a **complete star semiring**, the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:^{[17]}

A **Conway semiring** is a star semiring satisfying the sum-star and product-star equations:^{[7]}^{[26]}

An **iteration semiring** is a Conway semiring satisfying the Conway group axioms,^{[7]} associated by John Conway to groups in star-semirings.^{[28]}

The term **dioid** (for "double monoid") has been used to mean various types of semirings:

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 *hemirings*^{[32]} or *pre-semirings*.^{[33]} A further generalization are *left-pre-semirings*,^{[34]} which additionally do not require right-distributivity (or *right-pre-semirings*, which do not require left-distributivity).

Yet a further generalization are *near-semirings*: 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 numbers form a near-semiring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead.

In category theory, a *2-rig* is a category with functorial 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.