Nearly projective Boolean algebras
type: Item
instance of: scholarly article
title: Nearly projective Boolean algebras
zbMATH Open document ID: 0812.06007
author: Lutz Heindorf
author: Leonid B. Shapiro
published in: Lecture Notes in Mathematics
publication date: 1995-01-02T00:00:00Z
review text: 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 projec…
Mathematics Subject Classification ID: 06E05
Mathematics Subject Classification ID: 06-02
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 06E15
zbMATH DE Number: 707188
zbMATH Keywords: free Boolean algebras
zbMATH Keywords: survey
zbMATH Keywords: generalizations of projective Boolean algebras
zbMATH Keywords: openly generated Boolean spaces
zbMATH Keywords: Cohen algebras
zbMATH Keywords: consistency
reviewed by: J. Donald Monk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0094103
OpenAlex ID: W601141331
DOI: 10.1007/BFB0094103
description: scientific article; zbMATH DE number 707188