Abbreviation: ISRng$_{01}$

An 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