Constructions of Kleene lattices

We present an easy construction producing a Kleene lattice $\mathrm{K}=(K,\sqcup,\sqcap, {}^{\prime})$ from an arbitrary distributive lattice $\mathrm{L}=(L, \vee, \wedge)$ and a non-empty subset of $L$. We show that L can be embedded into K and compute $\vert K\vert$ under certain additional assumptions. We prove that every finite chain considered as a Kleene lattice can be represented in this way and that this construction preserves direct products. Moreover, we demonstrate that certain Kleene lattices that are ordinal sums of distributive lattices are representable. Finally, we prove that not every Kleene lattice is representable.