### Table of Contents

## Idempotent semirings with zero

Abbreviation: **ISRng**$_0$

### Definition

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

##### Morphisms

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:

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

### Examples

Example 1:

### Basic results

### Properties

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &2

f(3)= &12

f(4)= &129

f(5)= &1852

f(6)= &

\end{array}$