Abbreviation: ISRng$_0$

An \emph{idempotent semiring with zero} is a semirings with zero $\mathbf{S}=\langle S,\vee,0,\cdot \rangle$ such that $\vee$ is idempotent: $x\vee x=x$

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$

Example 1:

$\begin{array}{lr} f(1)= &1 f(2)= &2 f(3)= &12 f(4)= &129 f(5)= &1852 f(6)= & \end{array}$

Idempotent semirings with identity and zero

Idempotent semirings

Semirings with zero