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On the normal completion of a Boolean algebra
Authors:B Banaschewski  M Mahmoudi
Institution:a Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1
b Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran
Abstract:A familiar construction for a Boolean algebra A is its normal completionView the MathML source, given by its normal ideals or, equivalently, the intersections of its principal ideals, together with the embedding View the MathML source taking each element of A to its principal ideal. In the classical setting of Zermelo-Fraenkel set theory with Choice, View the MathML source is characterized in various ways; thus, it is the unique complete extension of A in which the image of A is join-dense, the unique essential completion of A, and the injective hull of A.Here, we are interested in characterizing the normal completion in the constructive context of an arbitrary topos. We show among other things that it is, even at this level, the unique join-dense, or alternatively, essential completion. En route, we investigate the functorial properties of View the MathML source and establish that it is the reflection of A, in the category of Boolean homomorphisms which preserve all existing joins, to the complete Boolean algebras. In this context, we make crucial use of the notion of a skeletal frame homomorphism.
Keywords:06E99  08B30  18A40  18D35
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