Abstract
AbstractA 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 $${\textbf{A}}$$
A
, namely its Dedekind-MacNeille completion $${{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})$$
DM
(
A
)
and a completion $$G({\textbf{A}})$$
G
(
A
)
which coincides with $${{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})$$
DM
(
A
)
provided $${\textbf{A}}$$
A
is finite. In particular we prove that if $${\textbf{A}}$$
A
is a Kleene poset then its extension $$G({\textbf{A}})$$
G
(
A
)
is also a Kleene lattice. If the subset X of principal order ideals of $${\textbf{A}}$$
A
is involution-closed and doubly dense in $$G({\textbf{A}})$$
G
(
A
)
then it generates $$G({\textbf{A}})$$
G
(
A
)
and it is isomorphic to $${\textbf{A}}$$
A
itself.
Funder
Austrian Science Fund
Grantová Agentura České Republiky
Publisher
Springer Science and Business Media LLC
Subject
History and Philosophy of Science,Logic
Reference9 articles.
1. Aglianò, P., and M. A. Marcos, Varieties of K-lattices, Fuzzy Sets and Systems 442: 222–248, 2022.
2. Busaniche, M., and R. Cignoli, The subvariety of commutative residuated lattices represented by twist-products, Algebra Universalis 71: 5–22, 2014.
3. Chajda, I., A note on pseudo-Kleene algebras, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 55: 39–45, 2016.
4. Chajda, I., and H. Langer, Kleene posets and pseudo-Kleene posets, Miskolc Mathematical Notes 23: 155–174, 2022.
5. Chajda, I., H. Langer, and J. Paseka, Constructions of Kleene lattices, in Proceeding of 2022 IEEE 52th International Symposium on Multiple-Valued Logic, Springer, 2022, pp. 88–93.