Abbreviation: KA

A \emph{Kleene algebra} is a structure $\mathbf{A}=\langle A,\vee ,0,\cdot ,1,^{\ast }\rangle$ of type $\langle 2,0,2,0,1\rangle$ such that $\langle A,\vee ,0,\cdot ,1\rangle$ is an idempotent semiring with identity and zero

$e\vee x\vee x^{\ast }x^{\ast }=x^{\ast }$

$xy\leq y\Longrightarrow x^{\ast }y=y$

$yx\leq y\Longrightarrow yx^{\ast }=y$

Let $\mathbf{A}$ and $\mathbf{B}$ be Kleene algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x^{\ast })=h(x)^{\ast }$, $h(0)=0$, and $h(1)=1$.

Example 1:

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

Action algebras

Kleene lattices

Idempotent semirings with identity and zero