On a poset algebra which is hereditarily but not canonically well generated

Let B be a superatomic Boolean algebra. B is well generated, if it has a well founded sublattice L such that L generates B. The free product of Boolean algebras B and C is denoted by B * C. If C is a chain then B(C) denotes the interval algebra over C. THEOREM 1: (a) Every Boolean subaJgebra of B(א₁) * B(א₀) is wellgenerated. (b) B(א₁) * B(א₁) contains a non well-generated Boolean subalgebra. Canonical well-generatedness is defined in the introduction. Recall that B(א₁) * B(א₀) is canonically well-generated, and thus well-generated. We prove the following result. THEOREM 2: B(א₁) * B(א₀) contains a non canonically well generated Boolean subalgebra. In contrast with Theorem 1(b), we have the following result. THEOREM 3: Let A = {a_α: α < א₁} ⊆ ℘(ω) be a strictly increasing sequence in the relation of almost containment. Let B be the subalgebra of ℘(ω) generated by {{n}: n ∈ א₀} ∪ A. Then B is superatomic, and B is not embeddable in a well-generated algebra.