Residuated Operators and Dedekind–MacNeille Completion

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F21%3A00118809" target="_blank" >RIV/00216224:14310/21:00118809 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/61989592:15310/21:73609450

Výsledek na webu

<a href="https://doi.org/10.1007/978-3-030-52163-9_5" target="_blank" >https://doi.org/10.1007/978-3-030-52163-9_5</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-030-52163-9_5" target="_blank" >10.1007/978-3-030-52163-9_5</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Residuated Operators and Dedekind–MacNeille Completion

Popis výsledku v původním jazyce

The concept of operator residuation for bounded posets with unary operation was introduced by the first two authors. It turns out that in some cases when these operators are transformed into lattice terms and the poset P is completed to its Dedekind–MacNeille completion DM(P) then the complete lattice DM(P) becomes a residuated lattice with respect to these transformed terms. It is shown that this holds in particular for Boolean posets and for relatively pseudocomplemented posets. A more complicated situation is with orthomodular and pseudo-orthomodular posets. We show which operators M (multiplication) and R (residuation) yield operator left-residuation in a pseudo-orthomodular poset P and if DM(P) is an orthomodular lattice then the transformed lattice terms circled dot and -&gt; form a left residuation in DM(P). However, it is a problem to determine when DM(P) is an orthomodular lattice. We get some classes of pseudo-orthomodular posets for which their Dedekind–MacNeille completion is an orthomodular lattice and we introduce the so-called strongly D-continuous pseudo-orthomodular posets. Finally we prove that, for a pseudo-orthomodular poset P, the Dedekind–MacNeille completion DM(P) is an orthomodular lattice if and only if P is strongly D-continuous.

Název v anglickém jazyce

Residuated Operators and Dedekind–MacNeille Completion

Popis výsledku anglicky

The concept of operator residuation for bounded posets with unary operation was introduced by the first two authors. It turns out that in some cases when these operators are transformed into lattice terms and the poset P is completed to its Dedekind–MacNeille completion DM(P) then the complete lattice DM(P) becomes a residuated lattice with respect to these transformed terms. It is shown that this holds in particular for Boolean posets and for relatively pseudocomplemented posets. A more complicated situation is with orthomodular and pseudo-orthomodular posets. We show which operators M (multiplication) and R (residuation) yield operator left-residuation in a pseudo-orthomodular poset P and if DM(P) is an orthomodular lattice then the transformed lattice terms circled dot and -&gt; form a left residuation in DM(P). However, it is a problem to determine when DM(P) is an orthomodular lattice. We get some classes of pseudo-orthomodular posets for which their Dedekind–MacNeille completion is an orthomodular lattice and we introduce the so-called strongly D-continuous pseudo-orthomodular posets. Finally we prove that, for a pseudo-orthomodular poset P, the Dedekind–MacNeille completion DM(P) is an orthomodular lattice if and only if P is strongly D-continuous.

Druh

C - Kapitola v odborné knize

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

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

Rok uplatnění

2021

Kód důvěrnosti údajů

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

Název knihy nebo sborníku

Algebraic Perspectives on Substructural Logics

ISBN

9783030521622

Počet stran výsledku

16

Strana od-do

57-72

Počet stran knihy

193

Název nakladatele

Springer

Místo vydání

Cham

Kód UT WoS kapitoly

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