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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References
Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories, the Joy of Cats. http://katmat.math.uni-bremen.de/acc
Aull, C.E., Thron, W.J.: Separation axioms between \(T_0\) and \(T_1\). Indag. Math. 24, 26–37 (1963)
Banaschewski, B., Bruns, G.: Injective hulls in the category of distributive lattices. J. Reine Angew. Math. 232, 102–109 (1968)
Banaschewski, B., Hager, A.: Essential completeness of archimedean \(\ell \)-groups with weak order unit. J. Pure Appl. Algebra 217, 915–926 (2013)
Banaschewski, B., Hager, A.: Essential completeness in categories of completely regular frames. Appl. Categ. Structures 21, 167–180 (2013)
Banaschewski, B., Pultr, A.: Variants of openness. Appl. Categ. Struct. 1, 181–190 (1993)
Balbes, R.: Projective and injective distributive lattices. Pac. J. Math. 21, 405–420 (1967)
Ball, R.N.: Distributive Cauchy lattices. Algebra Universalis 18, 134–174 (1984)
Ball, R.N., Picado, J., Pultr, A.: On an aspect of scatteredness in the point-free setting. Port. Math. 73, 139–152 (2016)
Gleason, A.M.: Projective topological spaces. Ill. J. Math. 2, 482–489 (1958)
Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)
Johnstone, P.T.: Stone Spaces. Cambridge Univ. Press, Cambridge (1982)
Picado, J., Pultr, A.: Frames and Locales, Topology without points. Frontiers in Mathematics, vol. 28. Springer, Basel (2012)
Picado, J., Pultr, A.: A Boolean extension of a frame, and a representation of discontinuity. Quaest. Math. 40, 1111–1125 (2017)
Picado, J., Pultr, A., Tozzi, A.: Joins of closed sublocales. Houston J. Math. (to appear)
Sikorski, R.: A theorem on extensions of homomorphisms. Ann. Soc. Pol. Math. 21, 332–335 (1948)
Simmons, H.: The lattice theoretic part of topological separation properties. Proc. Edinb. Math. Soc. 21(2), 41–48 (1978)
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Presented by J.B. Nation.
Dedicated to the memory of Bjarni Jónsson.
This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.
The authors gratefully acknowledge support from Project P202/12/G061 of the Grant Agency of the Czech Republic, and from the Department of Mathematics of the University of Denver.
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Ball, R.N., Pultr, A. Maximal essential extensions in the context of frames. Algebra Univers. 79, 32 (2018). https://doi.org/10.1007/s00012-018-0508-x
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DOI: https://doi.org/10.1007/s00012-018-0508-x