Document Type
Article
Publication Date
11-5-2021
Abstract
Exact meets in a distributive lattice are the meets Λiai such that for all b, (Λiai) V b = Λi(ai V 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 Sc(L) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet-generated by open sublocales.
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 φ : S --> C (where S is a join-semilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (φ-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 (ψ-exactness) connected with the lifts of ψ: S --> C with complemented values in more general distributive complete lattices C creating, again, frames of ψ-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 Sc(L) into it.
Recommended Citation
M.A. Moshier, J. Picado and A. Pultr, Some general aspects of exactness and strong exactness of meets. Topology and its Applications 309 (2022) 107906. https://doi.org/10.1016/j.topol.2021.107906
Peer Reviewed
1
Copyright
Elsevier
Comments
This is an author-produced PDF of an article (preprint) that later underwent peer review and was accepted for publication in Topology and its Applications, volume 309, in 2022. This article may not exactly replicate the final published version. The definitive publisher-authenticated version is available online at https://doi.org/10.1016/j.topol.2021.107906.