Centre for Mathematics, University of Coimbra

Some general aspects of exactness and strong exactness of meets (Preprint)

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Type:	Preprint
National /International:	International
Title:	Some general aspects of exactness and strong exactness of meets
Publication Date:	2021-02-12
Authors:	- M. Andrew Moshier - Jorge Picado - Ales Pultr
Abstract:	Exact meets in a distributive lattice are the meets ⋀_i a_i such that for all b, (⋀_i a_i)∨b = ⋀_i(a_i∨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 exacts resp. strongly exact lters are sublocales of the frame of upsets 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. 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 S_c(L) into it.
Institution:	DMUC 21-02
Online version:	http://www.mat.uc.pt...prints/eng_2021.html
Download:	Not available