Representability of Kleene Posets and Kleene Lattices

Result code in 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>

Alternative codes found

RIV/61989592:15310/24:73627583

Result on the web

<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>

Result language

angličtina

Original language name

Representability of Kleene Posets and Kleene Lattices

Original language description

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.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF20-09869L" target="_blank" >GF20-09869L: The many facets of orthomodularity</a><br>

Continuities

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

Publication year

2024

Confidentiality

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

Name of the periodical

Studia Logica

ISSN

0039-3215

e-ISSN

1572-8730

Volume of the periodical

112

Issue of the periodical within the volume

6

Country of publishing house

CH - SWITZERLAND

Number of pages

37

Pages from-to

1281-1317

UT code for WoS article

001116501800001

EID of the result in the Scopus database

2-s2.0-85178954026