Local and nonvanishing superatomic Boolean algebras with a distinguished dense subalgebra (Q695806)

If \(A\) is a Boolean algebra (BA) with a distinguished subalgebra (called just an algebra below), then the Frechet ideal of \(A\) is the ideal \(F(A)\) generated by the atoms. A subalgebra \(B\) of a BA \(A\) is called dense iff \(A\) is generated by \(B\cup F(A)\). The subalgebra has width \(n\) iff under any atom of \(B\) there are at most \(n\) atoms of \(A\), while every atom of \(A\) is below some atom of \(B\). \(K_n\) is the class of all superatomic BAs with a distinguished dense subalgebra of width \(n\). An algebra \(A\) is nonvanishing iff \(A\equiv B\times C\) implies that \(A\equiv B\) or \(A\equiv C\). We write \(A\leq B\) iff \(B\equiv A\times C\) for some \(C\). \(A\) is local iff there are only finitely many elementary nonequivalent nonvanishing algebras \(B\leq A\). A function \(w_A:N\rightarrow N\cup\{\infty\}\) is defined in terms of the usual Tarski invariants extended to algebras. One of the main theorems is that if \(A\) and \(B\) are local algebras in \(K_n\), then \(A\equiv B\) iff \(w_A=w_B\).