On a superatomic Boolean algebra which is not generated by a well-founded sublattice |
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Authors: | Uri Abraham Matatyahu Rubin Robert Bonnet |
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Institution: | (1) Department of Mathematics, Ben Gurion University of the Negev, 84105 Beer Sheva, Israel;(2) Laboratoire de Mathématiques, Université de Savoie, Le Bourget-du-Lac, France |
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Abstract: | Let b denote the unboundedness number of ωω. That is, b is the smallest cardinality of a subset
such that for everyg∈ωω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that
, and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ0, ℵ1, ℵ1, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ0, ℵ0, λ, 1〉, andB is not well-generated, then λ≥b. |
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