Some general aspects of exactness and strong exactness of meets

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10455253" target="_blank" >RIV/00216208:11320/22:10455253 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=eUlZ~jp3Gn" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=eUlZ~jp3Gn</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.topol.2021.107906" target="_blank" >10.1016/j.topol.2021.107906</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Some general aspects of exactness and strong exactness of meets

Popis výsledku v původním jazyce

Exact meets in a distributive lattice are the meets lambda(i) a(i) such that for all b, (lambda(i) a(i)) proves &amp; nbsp; b = lambda(i)(a(i )proves &amp; nbsp;b); strongly exact meets in a frame are preserved by all frame homomorphisms. Finite meets are, trivially, (strongly) exact; this naturally leads to the concepts of exact resp. strongly exact filters closed under all exact resp. strongly exact meets. In [2,12] it was shown that the subsets of all exact resp. strongly exact filters are sublocales of the frame of up-sets on a frame, hence frames themselves, and, somewhat surprisingly, that they are isomorphic to the useful frame S-c(L) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet generated by open sublocales.&amp; nbsp;In this paper we show that these are special instances of much more general facts. The latter concerns the free extension of join-semilattices to coframes; each coframe homomorphism lifting a general join-homomorphism phi: S &amp; nbsp;-&gt;&amp; nbsp;C (where S is a joinsemilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (phi-precise filters) and a closure operator on C (and - a minor point - any closure operator on C is thus obtained). The former case is slightly more involved: here we have an extension of the concept of exactness (0-exactness) connected with the lifts of 0: S &amp; nbsp;-&gt; C with complemented values in more general distributive complete lattices C creating, again, frames of 0-exact filters; as an application we learn that if such a C is join-generated (resp. meet-generated) by its complemented elements then it is a frame (resp. coframe) explaining, e.g., the coframe character of the lattice of sublocales, and the (seemingly paradoxical) embedding of the frame S-c(L) into it. (C)&amp; nbsp;2021 Elsevier B.V. All rights reserved.

Název v anglickém jazyce

Some general aspects of exactness and strong exactness of meets

Popis výsledku anglicky

Exact meets in a distributive lattice are the meets lambda(i) a(i) such that for all b, (lambda(i) a(i)) proves &amp; nbsp; b = lambda(i)(a(i )proves &amp; nbsp;b); strongly exact meets in a frame are preserved by all frame homomorphisms. Finite meets are, trivially, (strongly) exact; this naturally leads to the concepts of exact resp. strongly exact filters closed under all exact resp. strongly exact meets. In [2,12] it was shown that the subsets of all exact resp. strongly exact filters are sublocales of the frame of up-sets on a frame, hence frames themselves, and, somewhat surprisingly, that they are isomorphic to the useful frame S-c(L) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet generated by open sublocales.&amp; nbsp;In this paper we show that these are special instances of much more general facts. The latter concerns the free extension of join-semilattices to coframes; each coframe homomorphism lifting a general join-homomorphism phi: S &amp; nbsp;-&gt;&amp; nbsp;C (where S is a joinsemilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (phi-precise filters) and a closure operator on C (and - a minor point - any closure operator on C is thus obtained). The former case is slightly more involved: here we have an extension of the concept of exactness (0-exactness) connected with the lifts of 0: S &amp; nbsp;-&gt; C with complemented values in more general distributive complete lattices C creating, again, frames of 0-exact filters; as an application we learn that if such a C is join-generated (resp. meet-generated) by its complemented elements then it is a frame (resp. coframe) explaining, e.g., the coframe character of the lattice of sublocales, and the (seemingly paradoxical) embedding of the frame S-c(L) into it. (C)&amp; nbsp;2021 Elsevier B.V. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

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

Název periodika

Topology and its Applications

ISSN

0166-8641

e-ISSN

1879-3207

Svazek periodika

309

Číslo periodika v rámci svazku

March 2022

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

14

Strana od-do

107906

Kód UT WoS článku

000791838800001

EID výsledku v databázi Scopus

2-s2.0-85119016854