Abbreviation: ISRng$_{01}$

An \emph{idempotent semiring with identity and zero} is a semirings with identity and zero $\mathbf{S}=\langle S,\vee,0,\cdot,1 \rangle$ such that $\vee$ is idempotent: $x\vee x=x$

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$

Example 1:

$\begin{array}{lr} f(1)= & 1 f(2)= & 1 f(3)= & 3 f(4)= & 20 f(5)= & 149 f(6)= &1488 f(7)= &18554 \end{array}$

Kleene algebras

Idempotent semirings with zero

Idempotent semirings with identity

Semirings with identity and zero