Exact and Strongly Exact Filters
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Exact and Strongly Exact Filters M. A. Moshier1
· A. Pultr2 · A. L. Suarez3
Received: 8 December 2019 / Accepted: 14 July 2020 © Springer Nature B.V. 2020
Abstract A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame Filt E (L) somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters Filt sE (L). Keywords Frames · Locales · Sublocales open and closed · Exact meets and filters · Strongly exact meets and filters
1 Introduction The concept of an exact meet in a distributive lattice is fairly intuitive. Think of the lattice of open sets of a topological space; finite meets coincide with intersections, infinite meets typically do not. Those that do so exhibit special behavior, in particular they distribute over joins (that is, ( i ai ) ∨ b = i (ai ∨ b)), as the finite ones do. The history of exact meets goes back to MacNeille’s dissertation (1935) published in [10]; in Bruns and Lakser [6] they
Communicated by M. M. Clementino. The second author gratefully acknowledges support from KAM at MFF, Charles University, Prague and from CECAT at Chapman University, Orange.
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M. A. Moshier [email protected] A. Pultr [email protected] A. L. Suarez [email protected]
1
CECAT, Chapman University, Orange, CA 92688, USA
2
Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské nám. 24, 11800 Prague 1, Czech Republic
3
School of Computer Science, University of Birmingham, Birmingham, UK
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played a role in the study of injective hulls of meets semilattices. The fact that they generalize finite meets naturally leads to the notion of an exact filter, an up-set (increasing subset) closed under all exact meets. The system FiltE (L) of all exact filters in a frame L is a frame (a quotient of the frame U(L) of all up-sets of L). It turned out [2], rather surprisingly, that it was isomorphic to the frame Sc (L) of the joins of closed sublocales, an important device in studying various phenomena in point-free topology (like e.g. scatteredness, relation of subspaces and sublocales, modelling discontinuity). The concept of exact meet has a stronger modification, that of a strongly exact meet. It appeared (probably) first in 1993 in [16] and was later studied e.g. in [4] (see also [11]). The definition will be given below in 2.2, here it suffices to state that the strongly exact meets are precisely those that are preserved by all frame homomorphisms (recall that frame homomorphisms are explicitly requested to preserve finite meets; thus, again, strongly exact meets are a generalization of finite ones). And becau
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