Generalized free products (Q2717729)

The purpose of this paper is to define and study the generalized free product of Boolean algebras. The construction goes like this: NEWLINENEWLINENEWLINESuppose we're given a direct product \(A = \prod_{i \in I}A_i\) of Boolean algebras. Let \(B\) be a subalgebra of \(A\) with the property that if \(b \in B\) and \(c_i = b_i\) for all but finitely many \(i \in I\) then \(c \in B\) -- such a \(B\) is known as finitely closed. Let \(B^{\star} = \{b \in b: b_i \neq 0\) for all \(i\}\). Consider the coordinatewise partial order on \(B^{\star}\), and consider the poset topology generated by the open (and in fact regular open) sets \(\mathcal O_b = \{c \in B^{\star}: c \leq b\}\). Consider the algebra generated by the \(\mathcal O_b\)'s as a subalgebra of the regular open algebra on this topology. That's the \(B\)-generalized free product, denoted by \(\oplus^B_{i \in I}A_i\). NEWLINENEWLINENEWLINEThis complicated definition looks a little more natural given the following proposition: If \(B\) is the weak product of the \(A_i\)'s (i.e., the smallest finitely closed subalgebra possible), then the \(B\)-generalized free product is the free product. NEWLINENEWLINENEWLINEThe results in this paper include a characterization in terms of embeddings and a dense set of generators, and a number of results on cardinal invariants, especially when \(B\) is the full product \(\Pi_{i \in I}A_i\).