## 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:

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

Example 1:

### Properties

Classtype variety decidable undecidable no unbounded no no

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &12\\ f(4)= &129\\ f(5)= &1852\\ f(6)= &\\ \end{array}$