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:
| Classtype | quasivariety |
|---|---|
| Equational theory | decidable, PSPACE complete 1) |
| Quasiequational theory | undecidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | no |
| Congruence modular | no |
| Congruence meet-semidistributive | yes |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | |
| Definable principal congruences | |
| Equationally def. pr. cong. | |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &20
f(5)= &149
f(6)= &1488
\end{array}$