Commutative semiring

Definition

A commutative semiring is a semiring $S$ in which $ab=ba$ for all $a,b\in S$ . Thus both addition and multiplication are commutative, $1$ is part of the structure, and $0$ is multiplicatively absorbing.

Scope

Unless a page says otherwise, semirings in the tropical and hyperstructure parts of this corpus are commutative and unital. This convention makes their multiplicative monoids compatible with the commutative hyperrings and hyperfields used there.

Examples

Every commutative ring is a commutative semiring. The natural numbers, the Boolean semifield, and the max-plus tropical semifield are commutative semirings. A matrix semiring over a commutative semiring is generally not commutative when the matrix size exceeds one.

References

Jonathan S. Golan, Semirings and their Applications, Kluwer, 1999. Publisher DOI record. Relevant: Chapter 1.