### Table of Contents

## Idempotent semirings with identity and zero

Abbreviation: **ISRng**$_{01}$

### Definition

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: $x\vee x=x$

##### Morphisms

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:

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

### Examples

Example 1:

### Basic results

### Properties

### Finite members

$\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}$