A set S (containing a 0 and a 1) on which there are defined two dyadic operations that are denoted by + and • and that obey certain properties: the set S, regarded as a set with a zero on which the operation + is defined, is a monoid; the set S, regarded as a set with a unit on which • is defined, is a monoid; the operation + is commutative; the operation • is distributive over +. A semiring is said to be unitary if the operation • possesses a unit. A semiring is commutative if the operation • is commutative.
The set of polynomials in x whose coefficients are nonnegative integers constitutes an example of a semiring (which is not a ring), the two operations being addition and multiplication. Other uses of semirings occur in fuzzy logic. See also ring, closed semiring.