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Exact and strongly exact filters. (English) Zbl 1461.18010

There exists a well-know notion of frame, i.e., a complete lattice, in which finite meets distribute over arbitrary joins [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 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 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 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 (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 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 system of the fitted sublocales of \(L\) (a sublocale is fitted provided that it is an intersection of open sublocales, namely, sublocales of the form \(\{a\rightarrow b\,|\,b\in L\}\) for some \(a\in L\)), which addresses the question on representation of this system by filters considered in [R. N. Ball et al., Appl. Categ. Struct. 28, No. 4, 655–667 (2020; Zbl 1444.18018)]. The authors additionally show that the frame of exact filters \(\text{Filt}_{\text{E}}(L)\) is a sublocale of the frame of strongly exact filters \(\text{Filt}_{\text{sE}}(L)\).
The paper is well written and easy to read. It gives most of its required preliminaries, and will be of interest to the researchers studying point-free topology.

MSC:

18F70 Frames and locales, pointfree topology, Stone duality
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D20 Heyting algebras (lattice-theoretic aspects)
06D22 Frames, locales
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