Semiring - Knowlpedia

Definition

A semiring is a set $S$ with operations $+$ and $\cdot$ and elements $0,1$ such that $(S,+,0)$ is a commutative monoid, $(S,\cdot,1)$ is a monoid, multiplication distributes over addition on both sides, and $0s=s0=0$ for every $s\in S$ . The house convention includes a multiplicative identity but does not require $0\ne1$ or commutative multiplication.

Relation to rings

Every unital ring is a semiring after one forgets that additive inverses are part of the specified structure. Conversely, a semiring whose additive monoid is an abelian group is a unital ring. Thus semirings weaken the additive part of unital ring theory while keeping addition single-valued.

Examples and conventions

The natural numbers $\mathbb N$ , the nonnegative real numbers, and the endomorphisms of a commutative monoid under pointwise addition and composition are semirings. The last example can be noncommutative.

Some authors omit $1$ , allow noncommutative addition, or use rig to emphasize “ring without negatives.” Those are broader conventions than the one used here. In the rest of this subject, commutative semiring explicitly means that multiplication is also commutative.

References

Jonathan S. Golan, Semirings and their Applications, Kluwer, 1999. Publisher DOI record. Relevant: Chapter 1, semirings and their morphisms.
Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Idempotent (Asymptotic) Mathematics and the Representation Theory,” 2002. arXiv:math/0206025. Relevant: idempotent-semiring conventions and examples.