In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction ∨ {\displaystyle \lor } as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers N {\displaystyle \mathbb {N} } (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.
Terminology Some authors define semirings without the requirement for there to be a 0 {\displaystyle 0} or 1 {\displaystyle 1} . This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggesting that rigs are rings without negative elements. (Akin to using rng to mean a ring without a multiplicative identity.) The term dioid (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzmann in 1972 to denote a semiring. (It is alternatively sometimes used for naturally ordered semirings but the term was also used for idempotent subgroups by Baccelli et al. in 1992.)
Definition A semiring is a set R {\displaystyle R} equipped with two binary operations + {\displaystyle +} and ⋅ , {\displaystyle \cdot ,} called addition and multiplication, such that:
( R , + ) {\displaystyle (R,+)} is a commutative monoid with an identity element called 0 {\displaystyle 0} :
( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)}
0 + a = a {\displaystyle 0+a=a}
a + 0 = a {\displaystyle a+0=a}
a + b = b + a {\displaystyle a+b=b+a}
( R , ⋅ ) {\displaystyle (R,\,\cdot \,)} is a monoid with an identity element called 1 {\displaystyle 1} :
( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}
1 ⋅ a = a {\displaystyle 1\cdot a=a}
a ⋅ 1 = a {\displaystyle a\cdot 1=a}
Further, the following axioms tie to both operations:
Through multiplication, any element is left- and right-annihilated by the additive identity:
0 ⋅ a = 0 {\displaystyle 0\cdot a=0}
a ⋅ 0 = 0 {\displaystyle a\cdot 0=0}
Multiplication left- and right-distributes over addition:
a ⋅ ( b + c ) = ( a ⋅ b ) + ( a ⋅ c ) {\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}
( b + c ) ⋅ a = ( b ⋅ a ) + ( c ⋅ a ) {\displaystyle (b+c)\cdot a=(b\cdot a)+(c\cdot a)}
Notation The symbol ⋅ {\displaystyle \cdot } is usually omitted from the notation; that is, a ⋅ b {\displaystyle a\cdot b} is just written a b . {\displaystyle ab.}
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