Semiring

Remarks

The natural numbers $\N = \set{0, 1, 2, \dots}$ with the usual addition and multiplication form a commutative semiring. The four clauses are the familiar rules of arithmetic on $\N$. The tropical semiring is $\R \cup \set{\infty}$ with $\min$ as addition, ordinary $+$ as multiplication, $\infty$ as zero and $0$ as one. For instance, $a + \min(b, c) = \min(a + b, a + c)$ gives distributivity, and $a + \infty = \infty$ gives the absorbing clause. The positive integers $\set{1, 2, 3, \dots}$ with the usual operations are not a semiring: there is no additive identity, so clause 1 fails. A ring is exactly a semiring in which $(S, +)$ is a group; clause 4 then follows from the others. Some authors omit $0$ or $1$ from the definition, and some do not require clause 4.