Nearly projective Boolean algebras
This book is a self-contained survey, with full proofs, of various generalizations of projective Boolean algebras. Recall that a BA \(A\) is called projective if for any homomorphisms \(f : A \to B\) and \(g : C \to B\) with \(g\) onto, there is a homomorphism \(h : A \to C\) such that \(g \circ h = f\). This is an important concept of category theory, studied for many categories. The book studies in great detail several generalizations of projective BA's, of which we mention two. (1) A BA \(B\) is rc-filtered if one can associate with each element \(b \in B\) two finite sets \(U(b) \subseteq \{c \in B : b \leq c\}\) and \(L(b) \subseteq \{c \in B : c \leq b\}\) such that for all \(a,b \in B\), if \(a \leq b\) then \(U(a) \cap L(b) \neq \emptyset\). Several equivalent versions of this definition are given, one of which justifies the terminology (``rc'' for ``relatively complete''). Another equivalent gives the topological version: openly generated Boolean spaces, studied for more general spaces by Ščepin. The closure and non-closure properties of the class of rc-filtered BA's under algebraic operations are discussed, and a fairly complete description of the cardinal functions defined on these algebras is given. It is shown that if \(| B| = \aleph_ 1\), then \(B\) is isomorphic to the free BA on \(\aleph_ 1\) free generators iff \(B\) is a subalgebra of an rc-filtered BA and the Stone space of \(B\) is homogeneous. Several questions of Ščepin concerning these spaces are answered. Of course every projective algebra is rc-filtered. The theory of rc-filtered BA's is used to prove some results about the exponential of free BA's: for \(A\) free and of size \(\omega\) or \(\omega_ 1\), \(\text{Exp }A \cong A\), while for \(A\) free and of size \(> \omega_ 1\), \(\text{Exp }A\) is not even projective. (2) Two BA's \(A\) and \(B\) are co-absolute if they have isomorphic completions; this is the algebraic version of spaces being co-absolute. A BA \(A\) is called weakly projective if it is co-absolute with a projective BA. The ccc weakly projective BA's are exactly the Cohen algebras introduced by S. Koppelberg. Several equivalent definitions of weakly projective BA's are given. It is shown that every subalgebra of a projective BA is weakly projective. A BA is weakly projective iff its exponential is. There is an rc-filtered BA which is not weakly projective. The book closes with an appendix by S. Fuchino giving some consistency results about some of the notions of the book.
generalizations of projective Boolean algebras
scientific article; zbMATH DE number 707188