Power set modulo small, the singular of uncountable cofinality

Let μ be singular of uncountable cofinality. If μ > 2 ^cf(μ), we prove that in ℙ = ([μ]^μ, ⊇) as a forcing notion we have a natural complete embedding of Levy (N₀, μ⁺) (so ℙ collapses μ⁺ to N₀) and even Levy(N₀, U_Jκbd(μ)). The "natural" means that the forcing ({p ∈ [μ]^μ : p closed}, ⊇) is naturally embedded and is equivalent to the Levy algebra. Also if ℙ fails the χ-c.c. then it collapses χ to N₀ (and the parallel results for the case μ > N₀ is regular or of countable cofinality). Moreover we prove: for regular uncountable κ, there is a family P of b_κ partitions Ā = 〈A_α : α< κ〉 of κ such that for any A ∈ [κ] ^κ for some 〈 Aα : α < κ 〉 ∈ P we have α <κ ⇒ |A_α∩ A| = κ.