Semiring

A semiring is a set together with two Binary Operators satisfying the following conditions:

1. Additive associativity: For all , ,

2. Additive commutativity: For all , ,

3. Multiplicative associativity: For all , ,

4. Left and right distributivity: For all , and .
A semiring is therefore a commutative Semigroup under addition and a Semigroup under multiplication. A semiring can be empty.