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## Idempotent semirings with identity and zero

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

### Definition

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: $x\vee x=x$

##### 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

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