Idempotent semirings

Abbreviation: ISRng

Definition

An idempotent semiring is a semiring $\mathbf{S}=\langle S,\vee ,\cdot \rangle$ such that

$\vee$ is idempotent: $x\vee x=x$

Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be idempotent semirings. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:

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

Example 1:

Properties

Classtype Variety Decidable Undecidable No Unbounded No No

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &6\\ f(3)= &61\\ f(4)= &866\\ f(5)= &\\ f(6)= &\\ \end{array}$