Abbreviation: SRng

A \emph{semiring} is a structure $\mathbf{S}=\langle S,+,\cdot \rangle$ of type $\langle 2,2\rangle$ such that

$\langle S,\cdot\rangle$ is a semigroup

$\langle S,+\rangle$ is a commutative semigroup

$\cdot$ distributes over $+$: $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$

Let $\mathbf{S}$ and $\mathbf{T}$ be semirings. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$

Example 1:

$\begin{array}{lr} f(1)= &1 f(2)= &10 f(3)= &132 f(4)= &2341 f(5)= & f(6)= & \end{array}$

Idempotent semirings

Semirings with identity

Semirings with zero

Commutative semigroups