## Kleene algebras

Abbreviation: KA

### Definition

A 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

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

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

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

##### Morphisms

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:

### Properties

Classtype quasivariety decidable, PSPACE complete 1) undecidable undecidable no unbounded no no yes no no no

### Finite members

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

### References

1) L. J. Stockmeyer, A. R. Meyer, Word problems requiring exponential time: preliminary report, Fifth Annual ACM Symposium on Theory of Computing (Austin, Tex., 1973), Assoc. Comput. Mach., New York, 1973, 1–9 MRreviewZMATH