Exact and strongly exact filters
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 system of the fitted sublocales of \(L\) (a sublocale is \textit{fitted} provided that it is an intersection of \textit{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 [\textit{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.
scientific article; zbMATH DE number 7347068
scientific article