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## Idempotent semirings with zero

Abbreviation: **ISRng**$_0$

### Definition

An ** 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

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

### Subclasses

### Superclasses

### References

Trace: » idempotent_semirings_with_zero