Abstract
Exact meets in a distributive lattice are the meets lambda(i) a(i) such that for all b, (lambda(i) a(i)) proves & nbsp; b = lambda(i)(a(i )proves & nbsp;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 S-c(L) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet generated by open sublocales.& nbsp;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 phi: S & nbsp;->& nbsp;C (where S is a joinsemilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (phi-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 (0-exactness) connected with the lifts of 0: S & nbsp;-> C with complemented values in more general distributive complete lattices C creating, again, frames of 0-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 S-c(L) into it. (C)& nbsp;2021 Elsevier B.V. All rights reserved.