Table of Contents
Kleene lattices
Abbreviation: KLat
Definition
A \emph{Kleene lattice} is a structure $\mathbf{A}=\langle A,\vee ,\wedge ,0,\cdot ,1,^{\ast }\rangle $ of type $\langle 2,2,0,2,0,1\rangle $ such that
$\langle A,\vee ,0,\cdot ,1,^{\ast }\rangle $ is a Kleene algebra
$\langle A,\vee ,\wedge \rangle $ is a lattice
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be Kleene lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(x\cdot y)=h(x)\cdot h(y)$, $h(x^{\ast })=h(x)^{\ast }$, $h(0)=0$, $h(1)=1$
Examples
Example 1:
Basic results
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &16
f(5)= &149
f(6)= &1488
\end{array}$
