### Table of Contents

## Kleene algebras

Abbreviation: **KA**

### Definition

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

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

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

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

##### 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 $h(1)=1$.

### Examples

Example 1:

### Basic results

### Properties

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 |

### 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}$