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.

克林格是具有反调对合并满足所谓的正规性条件的分配格。这些格子是由 J. A. Kalman 提出的。我们也将这个概念扩展到具有反音对合的偏序集。在我们最近的论文（Chajda、Länger 和 Paseka，发表于：2022 年 IEEE 第 52 届多值逻辑国际研讨会论文集，Springer，2022 年）中，我们展示了如何从给定的分配格或偏序集构造此类 Kleene 格或 Kleene 偏序集，以及分别使用所谓的扭曲积结构来构成该晶格或偏序集的固定元素。我们通过考虑固定子集而不是固定元素来扩展克林格和克林偏序集的构造。此外，我们表明，在某些情况下，该生成偏序集可以嵌入到生成的 Kleene 偏序集中。我们研究克莱尼偏序集何时可以由通过上述构造获得的克莱尼偏序集来表示的问题。我们证明了可表示 Kleene 偏集的直积再次是可表示的，因此有限链的直积是可表示的。这通常不适用于子直接产品，但我们展示了一些它适用的示例。我们提出了一大类可表示和不可表示的克莱尼偏序集。最后，我们研究了分配偏序集\({\textbf{A}}\)的两种扩展，即其 Dedekind-MacNeille 补全\({{\,\mathrm{\textbf{DM}}\,}}( {\textbf{A}})\)和补全\(G({\textbf{A}})\)与\({{\,\mathrm{\textbf{DM}}\,}}( {\textbf{A}})\)假设\({\textbf{A}}\)是有限的。特别地，我们证明如果\({\textbf{A}}\)是一个 Kleene 偏集，那么它的扩展\(G({\textbf{A}})\)也是一个 Kleene 格子。如果\({\textbf{A}}\)的主阶理想的子集X在\(G({\textbf{A}})\)中是对合闭且双稠密的，那么它会生成\(G({ \textbf{A}})\)并且它与\({\textbf{A}}\)本身同构。