Limitations on representing P(X) as a union of proper subalgebras

For every integer μ ≥ 3, there exists a function f_μ: N₊ → N₊ such that the following holds: (1) f_μ(k) = 2k − μ for k large enough; (2) if A is a finite nonempty collection of subalgebras of P(X) such that ⊓ B is not f_μ (#(B))-saturated, for all nonempty B ⊆ A, then ⊔ A ≠ P(X).