### Table of Contents

## Kleene logic algebras

Abbreviation: **KLA**

### Definition

A \emph{Kleene logic algebra} is a De Morgan algebra $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\neg\rangle $ that satisfies

$x\wedge \neg x\le y\vee \neg y$.

Remark: Also called Kleene algebras, but this name more commonly refers to the algebraic models of regular languages.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be Kleene logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

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

### Examples

Example 1: Let $\{0<a<1\}$ be the 3-element lattice with $0'=1,a'=a,b'=b$.

### Basic results

The algebra in Example 1 generates the variety of Kleene logic algebras

### Properties

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &1

f(3)= &1

f(4)= &2

f(5)= &1

f(6)= &3

f(7)= &2

f(8)= &6

f(9)= &4

f(10)= &10

\end{array}$