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α -cut-complete Boolean algebras TeX Export

Algebra Universalis, Vol. 39, No. 1. (2 July 1998), pp. 57-70.

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Let A be a Boolean algebra, and $ α $ an infinite cardinal number or the symbol $ ∞ $. An $ α $-cut in A is an ordered pair (F,H) of subsets of A, each of power $ ≤ α $, with $ F ≤ H $ elementwise, with 0 as the meet of differences $ h - f (h ∈ H, f ∈ F) $. A is called $ α $-cut-complete if for each $ α $-cut (F,H) there is $ a ∈ A $ with $ F ≤ a ≤ H $ elementwise. We describe the simply-constructed $ α $-cut-completion $ A^α $, show that $ α $-cut-completeness solves a natural $ α $-injectivity problem, determine when $ A^α $ is the $ α $-completion, or the completion, and interpret most of that topologically in Stone spaces. Oddly, these considerations seem novel in Boolean algebras, while for lattice-ordered groups and vector lattices, and dually for topological spaces, the analogous theory, especially for $ α = ω_1 $, has received considerable study.


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