Exact and strongly exact filters
type: Item
instance of: scholarly article
title: Exact and strongly exact filters
zbMATH Open document ID: 1461.18010
published in: Applied Categorical Structures
publication date: 2021-05-11T00:00:00Z
full work available at URL: https://digitalcommons.chapman.edu/scs_articles/682
review text: There exists a well-know notion of \textit{frame}, i.e., a complete lattice, in which finite meets distribute over arbitrary joins~[\textit{P. T. Johnstone}, Stone spaces. Cambridge: Cambridge University Press (1982; Zbl 0499.54001)]. The paper considers the notion of (strongly) exact meet and its related concept of (strongly) exact filter in frames. A meet \(\bigwedge A\) in a frame \(L\) is said to be \textit{exact} provided that it distributes over joins, i.e., \((\bigwedge A)\vee b=\bigwedge_{a\in A}(a\vee b)\) for every \(b\in L\). Moreover, \(\bigwedge A\) is said to be \textit{strongly exact} provided that it is preserved by every frame homomorphism \(h\), i.e., \(h(\bigwedge A)=\bigwedge_{a\in A}h(a)\) (every frame homomorphism preserves finite meets and arbitrary joins). The following implications for meets are valid in frames: ``finite'' \(\Rightarrow\) ``strongly exact'' \(\Rightarrow\) ``exact''. The classical notion of \textit{filter} in a frame as an up-set (a subset \(A\) of a frame \(L\) such that for every \(a\in A\) and every \(b\in L\), \(a\leqslant b\) implies \(b\in A\)) closed under finite meets can thus be easily extended to \textit{(strongly) exact filters}, namely, as up-sets closed under (strongly) exact meets. The paper provides a characterization of the sets of exact and strongly exact filters of a frame \(L\), denoted \(\text{Filt}_{\text{E}}(L)\) and \(\text{Filt}_{\text{sE}}(L)\), respectively. One of the main characterization tools makes the notion of frame sublocale. A subset \(S\) of a frame \(L\) is a \textit{sublocale} provided that it fulfills the following two properties: (1) if \(M\subseteq S\), then \(\bigwedge M\in S\); (2) if \(s\in S\) and \(a\in L\), then \(a\rightarrow s\in S\), where the operation \(\cdot\rightarrow\cdot\) is defined by \(a\wedge b\leqslant c\) iff \(a\leqslant b\rightarrow c\) for every \(a,b,c,\in L\). The authors show, in particular, that \(\text{Filt}_{\text{sE}}(L)\) is naturally isomorphic to the s…
reviewed by: Sergejs Solovjovs
Mathematics Subject Classification ID: 18F70
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 06D22
zbMATH DE Number: 7347068
zbMATH Keywords: congruence
zbMATH Keywords: frame
zbMATH Keywords: Galois adjoint maps
zbMATH Keywords: Heyting structure
zbMATH Keywords: locale
zbMATH Keywords: localic map
zbMATH Keywords: nucleus
zbMATH Keywords: sober space
zbMATH Keywords: (strongly) exact filter
zbMATH Keywords: (strongly) exact meet
zbMATH Keywords: sublocale
author: M. Andrew Moshier
author: A. Pultr
author: Anna Laura Suarez
MaRDI profile type: Publication
OpenAlex ID: W3044861675
cites work: Q3844856
cites work: Exact filters and joins of closed sublocales
cites work: Distributive Cauchy lattices
cites work: Notes on exact meets and joins.
cites work: On an aspect of scatteredness in the point-free setting
cites work: Injective Hulls of Semilattices
cites work: The other closure and complete sublocales
cites work: Atomless Parts of Spaces.
cites work: Q3965241
cites work: Extensions of Partially Ordered Sets
cites work: Free meets and atomic assemblies of frames
cites work: Frames and Locales
cites work: A Boolean extension of a frame and a representation of discontinuity
cites work: Q5224415
cites work: Sublocale lattices
DOI: 10.1007/S10485-020-09602-0
Recommended article: Q5769243
Recommended article: Q4998511
Recommended article: Finite-dimensional filters
Recommended article: Perfect filtering and double disjointness
Recommended article: Q4267646
Recommended article: The maximality of filters
Recommended article: Q4955456
Recommended article: Minimal Extrapolations of Filters
Recommended article: Q3733735
description: scientific article; zbMATH DE number 7347068