Cellularity of free products of Boolean algebras (or topologies)

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, θ = (2^cf(μ))⁺ and 2^μ = μ⁺ then here are Boolean algebras double-struck B sign₁, double-struck B sign₂ such that c(double-struck B sign₁) = μ, c(double-struck B sign₂) < θ but c(double-struck B sign₁ * double-struck B sign₂) = μ⁺. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if double-struck B sign is a ccc Boolean algebra and μ^{(hebrew letter gimel)ω} ≤ λ = cf(λ) ≤ 2^μ then double-struck B sign satisfies the λ-Knaster condition (using the "revised GCH theorem").