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 \({\mathrm{Filt}}_{{\textsf {E}}}(L)\) 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 \({\mathrm{Filt}}_{{\textsf {sE}}}(L)\).
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Notes
This may sound odd but it makes good sense; if L happens to have points, they are sublocales of the form \(\{a,1\}\) with prime \(a\ne 1\). So \({\textsf {O}}\) is even smaller than a point.
This can be rewritten in first order formulas as follows:
$$\begin{aligned} a \nleq b\ \implies \exists c,\ a \vee c = 1 \ne b \vee c, \quad \end{aligned}$$(sfit)and
$$\begin{aligned} a \nleq b \implies \exists c,\ a \vee c = 1\ \text {and}\ c \!\rightarrow \!b \ne b. \end{aligned}$$(fit)
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Communicated by M. M. Clementino.
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The second author gratefully acknowledges support from KAM at MFF, Charles University, Prague and from CECAT at Chapman University, Orange.
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Moshier, M.A., Pultr, A. & Suarez, A.L. Exact and Strongly Exact Filters. Appl Categor Struct 28, 907–920 (2020). https://doi.org/10.1007/s10485-020-09602-0
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DOI: https://doi.org/10.1007/s10485-020-09602-0