Semifield - Knowlpedia

Definition

A semifield is a commutative semiring $S$ with $0\ne1$ such that $S^\times=S\setminus\{0\}$ . Equivalently, the nonzero elements form an abelian group under multiplication. Additive inverses are not required.

Examples and non-examples

Every field is a semifield. The nonnegative real numbers, the Boolean semifield, and tropical semifields are semifields that are not fields. The natural numbers are not a semifield because, for example, $2$ has no multiplicative inverse in $\mathbb N$ .

Convention warning

In some literature semifield means a possibly noncommutative division semiring; in finite geometry it can instead mean a nonassociative division algebra. Neither broader usage is intended here: multiplication is commutative and associative. A hyperfield is a different object because its addition may be multivalued.

References

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