Some general aspects of exactness and strong exactness of meets

Abstract

Exact meets in a distributive lattice are the meets ⋀iai such that for all b, (⋀iai)∨b=⋀i(ai∨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.