Definition (Semiring)

A semiring is a triple $(R, +, \cdot)$ where

• $(R,+)$ is a semigroup
• $(R,\cdot)$ is a semigroup
• Distributivity: $$\begin{align*} a\cdot(b+c)=ab+ac\\ (b+c)\cdot a=ba+ca \end{align*}$$