Abstract

MARTINEZ, Juan Carlos. A direct proof of a theorem of Jech and Shelah on PCF algebras. Rev.colomb.mat. [online]. 2018, vol.52, n.2, pp.131-137. ISSN 0034-7426.  https://doi.org/10.15446/recolma.v52n2.77153.

By using an argument based on the structure of the locally compact scattered spaces, we prove in a direct way the following result shown by Jech and Shelah: there is a family {Bα: α < ω1} of subsets of ω1 such that the following conditions are satisfied:

(a) max B α - α,

(b) if α ∈ B β then Bα ⊆ B β,

(c) if δ ≤ α and δ is a limit ordinal then Bα ∩ δ is not in the ideal generated by the sets Bβ, β < α, and by the bounded subsets of δ,

(d) there is a partition {An: n ∈ ω} of ω1 such that for every α and every n, B α ∩ A n is finite.

Keywords : PCF theory; locally compact scattered space.

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