Representability of Kleene Posets and Kleene Lattices

A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J. A. Kalman. We extended this concept also for posets with an antitone involution. In our recent paper (Chajda, Länger and Paseka, in: Proceeding of 2022 IEEE 52th International Symposium on Multiple-Valued Logic, Springer, 2022), we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and a fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element. Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset. We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset \({\textbf{A}}\), namely its Dedekind-MacNeille completion \({{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})\) and a completion \(G({\textbf{A}})\) which coincides with \({{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})\) provided \({\textbf{A}}\) is finite. In particular we prove that if \({\textbf{A}}\) is a Kleene poset then its extension \(G({\textbf{A}})\) is also a Kleene lattice. If the subset X of principal order ideals of \({\textbf{A}}\) is involution-closed and doubly dense in \(G({\textbf{A}})\) then it generates \(G({\textbf{A}})\) and it is isomorphic to \({\textbf{A}}\) itself.