Representability of Kleene Posets and Kleene Lattices

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F24%3A00139677" target="_blank" >RIV/00216224:14310/24:00139677 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/61989592:15310/24:73627583

Výsledek na webu

<a href="https://link.springer.com/article/10.1007/s11225-023-10080-3" target="_blank" >https://link.springer.com/article/10.1007/s11225-023-10080-3</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11225-023-10080-3" target="_blank" >10.1007/s11225-023-10080-3</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Representability of Kleene Posets and Kleene Lattices

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Representability of Kleene Posets and Kleene Lattices

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF20-09869L" target="_blank" >GF20-09869L: Ortomodularita z různých pohledů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Studia Logica

ISSN

0039-3215

e-ISSN

1572-8730

Svazek periodika

112

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

37

Strana od-do

1281-1317

Kód UT WoS článku

001116501800001

EID výsledku v databázi Scopus

2-s2.0-85178954026