1. Additive associativity: For all a, b, c in S, (a + b) + c = a + (b + c),
2. Additive commutativity: For all a, b in S, a + b = b + a,
3. Multiplicative associativity: For all a, b, c in S, (a * b) * c = a * (b * c),
4. Left and right distributivity: For all a, b, c in S, a * (b + c) = (a * b) + (a * c) and (b + c) * a = (b * a) + (c * a).
A semiring is therefore a commutative semigroup under addition and a semigroup under multiplication. A semiring can be empty.
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