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
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &1
f(6)= &3
f(7)= &2
f(8)= &6
f(9)= &4
f(10)= &10
\end{array}$
