Kleene logic algebras

Abbreviation: KLA

Definition

A 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

Classtype Variety Decidable Yes Yes Yes Yes 3

Finite members

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