Lindelof tightness and the Dedekind-MacNeille completion of a regular sigma-frame

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10368629" target="_blank" >RIV/00216208:11320/17:10368629 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.2989/16073606.2017.1288665" target="_blank" >http://dx.doi.org/10.2989/16073606.2017.1288665</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.2989/16073606.2017.1288665" target="_blank" >10.2989/16073606.2017.1288665</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Lindelof tightness and the Dedekind-MacNeille completion of a regular sigma-frame

Popis výsledku v původním jazyce

Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which subobjects of an object in the subcategory reflect to it; those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis of lambda-tightness, tightness with respect to the localic Lindeldof reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular sigma-frame A is a frame. Moreover, the embedding of A in its normal completion is the Bruns-Lakser injective hull of A in the category of meet semilattices and semilattice homomorphisms.Since every regular sigma-frame is the cozero part of a regular Lindeldof frame, this result points towards lambda-tightness. For any regular Lindeldof frame L, the normal completion of Coz L embeds in L as the sublocale generated by Coz L. Although this completion is clearly contained in every sublocale having the same cozero part as L, we show by example that its cozero part need not be the same as the cozero part as L. We prove that a sublocale S is lambda-tight in L iff S has the same cozero part as L. The aforementioned counterexample shows that the completion of Coz L is not always -tight in L; on the other hand, we present a large class of locales for which this is the case.

Název v anglickém jazyce

Lindelof tightness and the Dedekind-MacNeille completion of a regular sigma-frame

Popis výsledku anglicky

Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which subobjects of an object in the subcategory reflect to it; those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis of lambda-tightness, tightness with respect to the localic Lindeldof reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular sigma-frame A is a frame. Moreover, the embedding of A in its normal completion is the Bruns-Lakser injective hull of A in the category of meet semilattices and semilattice homomorphisms.Since every regular sigma-frame is the cozero part of a regular Lindeldof frame, this result points towards lambda-tightness. For any regular Lindeldof frame L, the normal completion of Coz L embeds in L as the sublocale generated by Coz L. Although this completion is clearly contained in every sublocale having the same cozero part as L, we show by example that its cozero part need not be the same as the cozero part as L. We prove that a sublocale S is lambda-tight in L iff S has the same cozero part as L. The aforementioned counterexample shows that the completion of Coz L is not always -tight in L; on the other hand, we present a large class of locales for which this is the case.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Centrum excelence - Institut teoretické informatiky (CE-ITI)</a><br>

Návaznosti

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

Rok uplatnění

2017

Kód důvěrnosti údajů

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

Název periodika

Quaestiones Mathematicae

ISSN

1607-3606

e-ISSN

—

Svazek periodika

40

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

ZA - Jihoafrická republika

Počet stran výsledku

16

Strana od-do

347-362

Kód UT WoS článku

000401713200005

EID výsledku v databázi Scopus

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