Semiring

A semiring is a set together with two Binary Operators $S(+,*)$ satisfying the following conditions:

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.

See also Binary Operator, Ring, Ringoid, Semigroup

References

Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.