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- W4286851350 abstract "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 [5], 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 A, namely its Dedekind-MacNeille completion DM(A) and a completion G(A) which coincides with DM(A) provided A is finite. In particular we prove that if A is a Kleene poset then its extension G(A) is also a Kleene lattice. If the subset X of principal order ideals of A is involutionclosed and doubly dense in G(A) then it generates G(A) and it is isomorphic to A itself." @default.
- W4286851350 created "2022-07-25" @default.
- W4286851350 creator A5005373819 @default.
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- W4286851350 date "2021-11-21" @default.
- W4286851350 modified "2023-09-24" @default.
- W4286851350 title "Representability of Kleene posets and Kleene lattices" @default.
- W4286851350 doi "https://doi.org/10.48550/arxiv.2111.10823" @default.
- W4286851350 hasPublicationYear "2021" @default.
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