Some classes of projective Boolean algebras

Abstract

Denote by P the class of projective Boolean algebras and by K1 the class of those Boolean algebras which are isomorphic to a product of a finite set of algebras each of which is a coproduct of countable algebras. It is well-known that K 1 = P; answering a question of Halmos, we showed in [3] that Kl is a proper subclass of P. This result is improved in this paper in several ways. We define a notion of weak product of Boolean algebras and show that P is closed with respect to countable weak products. Let K (resp. L) be the smallest class of Boolean algebras containing every countable algebra and being closed with respect to coproducts and finite products (resp. coproducts, finite products and countable weak products). We shall show that K~ is a proper subclass of K, K is a proper subclass of L and L is a proper subclass of P. The algebra proving L :k P is the same as constructed for the counterexample in [3]; the algebras working for K1 4: K and K ~: L are much simpler examples proving K 1 # P. After some preliminaries in Chapter 1, we show K 1 :~ K in Chapter 2. Weak products are introduced in Chapter 3. In Chapter 4, we prove that for every A e K there is a B ~ K1 such that A and B have isomorphic completions; this yields K + L . Finally, L~: P is derived in Chapter 5.