Maximal essential extensions in the context of frames |
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Authors: | Richard N. Ball Aleš Pultr |
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Affiliation: | 1.Department of Mathematics,University of Denver,Denver,USA;2.Department of Applied Mathematics and CE-ITI, MFF,Charles University,Prague 1,Czech Republic |
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Abstract: | We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through any other essential extension. This extension can be realized as the embedding (L rightarrow mathcal {N}(L) rightarrow mathcal {B}mathcal {N}(L)), where (L rightarrow mathcal {N}(L)) is the familiar embedding of L into its congruence frame (mathcal {N}(L)), and (mathcal {N}(L) rightarrow mathcal {B}mathcal {N}(L)) is the Booleanization of (mathcal {N}(L)). Finally, we show that for subfit frames the extension can also be realized as the embedding (L rightarrow {{mathrm{S}}}_mathfrak {c}(L)) of L into its complete Boolean algebra ({{mathrm{S}}}_mathfrak {c}(L)) of sublocales which are joins of closed sublocales. |
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