Representability of Kleene Posets and Kleene Lattices

J 2024

Representability of Kleene Posets and Kleene Lattices

CHAJDA, Ivan; Helmut LÄNGER a Jan PASEKA

Základní údaje

Originální název

Representability of Kleene Posets and Kleene Lattices

Autoři

CHAJDA, Ivan; Helmut LÄNGER a Jan PASEKA

Vydání

Studia Logica, Springer, 2024, 0039-3215

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Švýcarsko

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 0.600

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/24:00139677

Organizační jednotka

Přírodovědecká fakulta

Klíčová slova anglicky

Kleene lattice; Normality condition; Kleene poset; Pseudo-Kleene poset; Representable Kleene lattice; Embedding; Twist-product; Dedekind-MacNeille completion

Štítky

14311010, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 8. 1. 2025 09:16, Mgr. Marie Novosadová Šípková, DiS.

Anotace

V originále

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, Langer 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 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 involution-closed and doubly dense in G(A) then it generates G(A) and it is isomorphic to A itself.

Návaznosti

GF20-09869L, projekt VaV

Název: Ortomodularita z různých pohledů

Investor: Grantová agentura ČR, Ortomodularita z různých pohledů, Partnerská agentura (Rakousko)

Citovat

CHAJDA, Ivan; Helmut LÄNGER a Jan PASEKA. Representability of Kleene Posets and Kleene Lattices. Studia Logica. Springer, 2024, roč. 112, č. 6, s. 1281-1317. ISSN 0039-3215. Dostupné z: https://doi.org/10.1007/s11225-023-10080-3.

@article{2465892,
   author = {Chajda, Ivan and Länger, Helmut and Paseka, Jan},
   article_number = {6},
   doi = {https://doi.org/10.1007/s11225-023-10080-3},
   keywords = {Kleene lattice; Normality condition; Kleene poset; Pseudo-Kleene poset; Representable Kleene lattice; Embedding; Twist-product; Dedekind-MacNeille completion},
   language = {eng},
   issn = {0039-3215},
   journal = {Studia Logica},
   title = {Representability of Kleene Posets and Kleene Lattices},
   url = {https://link.springer.com/article/10.1007/s11225-023-10080-3},
   volume = {112},
   year = {2024}
}

TY  - JOUR
ID  - 2465892
AU  - Chajda, Ivan - Länger, Helmut - Paseka, Jan
PY  - 2024
TI  - Representability of Kleene Posets and Kleene Lattices
JF  - Studia Logica
VL  - 112
IS  - 6
SP  - 1281-1317
EP  - 1281-1317
PB  - Springer
SN  - 00393215
KW  - Kleene lattice
KW  - Normality condition
KW  - Kleene poset
KW  - Pseudo-Kleene poset
KW  - Representable Kleene lattice
KW  - Embedding
KW  - Twist-product
KW  - Dedekind-MacNeille completion
UR  - https://link.springer.com/article/10.1007/s11225-023-10080-3
N2  - 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, Langer 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 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 involution-closed and doubly dense in G(A) then it generates G(A) and it is isomorphic to A itself.
ER  -

CHAJDA, Ivan; Helmut LÄNGER a Jan PASEKA. Representability of Kleene Posets and Kleene Lattices. \textit{Studia Logica}. Springer, 2024, roč.~112, č.~6, s.~1281-1317. ISSN~0039-3215. Dostupné z: https://doi.org/10.1007/s11225-023-10080-3.

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