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

LetB be a superatomic Boolean algebra.B is well generated, if it has a well founded sublatticeL such thatL generatesB. The free product of Boolean algebrasB andC is denoted byB *C. IfC is a chain thenB(C) denotes the interval algebra overC.Theorem 1: (a)Every Boolean subalgebra of B(ℵ1) *B(ℵ0)is well-generated.(b)B(ℵ1) *B(ℵ1)contains a non well-generated Boolean subalgebra.Canonical well-generatedness is defined in the introduction. Recall thatB(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 ={ɑ:α