## 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

### Citations:

Zbl 0499.54001; Zbl 1444.18018
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### References:

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