Abbreviation: ISRng$_1$

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

$\vee$ is idempotent: $x\vee x=x$

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

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

Example 1:

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

Idempotent semirings with identity and zero

Idempotent semirings

Semirings with identity