Abbreviation: KLat

A 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

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:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(x\cdot y)=h(x)\cdot h(y)$, $h(x^{\ast })=h(x)^{\ast }$, $h(0)=0$, $h(1)=1$

Example 1:

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &16\\ f(5)= &149\\ f(6)= &1488\\ \end{array}$

Action lattices

Kleene algebras

Multiplicative lattices