Semirings

Why

We abstract the properties of the natural numbers under natural addition and multiplication.

Definition

A semiring $(S, +, \cdot )$ satisfies all the properties of a ring (see Rings) except that addition $+$ need not have additive inverses.

Examples

Set of natural numbers. The set $(\N , +, \cdot )$ where $+$ and $\cdot $ denote natural addition and multiplication respectively is a semiring.

Nonnegative real numbers with max and multiplication Notice that

\[ \max(a, b) = \max(b, a) \quad \text{for all } a, b \in \R \]

\[ \max(a, \max(b, c)) = \max(\max(a, b), c) \quad \text{for all } a, b, c \in \R \]

So $\max: \R ^2 \to \R $ is a commutative and associative operation. The identity is $0$, $\max(a, 0) = a$ for all $a \in \R _+$. Notice that there is no inverse element. Of course, $\cdot $ is associative and has identities.