# Semi-ring

2010 Mathematics Subject Classification: Primary: 16Y60 [MSN][ZBL]

A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the distributive laws $$(a+b) \cdot c = a\cdot c + b \cdot c$$ and $$a \cdot (b+c) = a\cdot b + a\cdot c \ .$$ In most cases one also assumes that the addition is commutative and that there exists a zero element $0$ such that $a + 0 = a$ for every $a \in S$. The most important classes of semi-rings are rings and distributive lattices. If there is a multiplicative unit element 1, the two classes are combined by the condition $$\forall x \, \exists y \ x+y=1 \ .$$ The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.

The term "exotic" semi-rings has been used to describe subsets of the real numbers with $\min$ or $\max$ as ${+}$ and addition as ${\star}$. These are thus idempotent semi-rings. Examples include the tropical semiring on $\mathbf{N} \cup \{\infty\}$ with operations ${\min},\, +$.
An additive zero in a semiring $S$ is an element $a$ such that $a+x = x+a = x$ for all $x$; a multiplicative zero is an element $m$ such that $m \cdot x = x \cdot m = m$ for all $x$. A double zero is an element which is both an additive zero and a multiplicative zero.
If the additive semigroup of a semiring $S$ is commutative and satisfies the cancellative property $a + c = b + c \Rightarrow a = b$ for all $c$, then the additive semigroup embeds in its Grothendieck group $R$ and the multiplication $\cdot$ extends to $R$, giving it a ring structure: the Grothendieck ring of $S$. The Grothendieck ring of a finite group $G$ over a field $K$ is the ring constructed in this way from the semiring of isomorphism classes of modules over the group ring $K[G]$ with direct sum and tensor product as the operations.