Small tree algebras with nontree subalgebras
type: Item
instance of: scholarly article
title: Small tree algebras with nontree subalgebras
zbMATH Open document ID: 0795.06014
DOI: 10.1016/0166-8641(93)90149-8
author: Eric K. van Douwen
published in: Topology and its Applications
publication date: 1993-11-15T00:00:00Z
review text: Given a partial order \(P\) we can consider the Boolean algebra generated by the left-closed cones (\(\{s: s\geq t\}\) for all \(t\in P\)). If \(P\) is a tree (each \(\{s: s\leq t\}\) is well-ordered) then we call this algebra a tree algebra. In particular, the interval algebra on an ordinal is a tree algebra. Define a family of sets to be non-Archimedean iff any two elements are comparable (via subset) or disjoint. This paper contains three theorems about tree algebras and non-Archimedean families. Theorem. The interval algebra on \(\omega_ 1\cdot 2\) has a nontree subalgebra which is an interval algebra, and a nontree subalgebra which is not an interval algebra. These algebras are the clopen algebras of the two quotient spaces. In the first quotient space we take two disjoint copies of the space \(\omega_ 1+ 1\) and identify the two copies of \(\omega_ 1\) (call this point \(\infty\)); in the second space we add a copy of \(\omega+1\) and identify its limit point with \(\infty\). On the other hand, \(\omega_ 1\cdot 2\) is the smallest ordinal, where this is possible: Theorem. Every subalgebra of the interval algebra on \(\omega_ 1\) is a tree algebra. The proof of this theorem relies on a characterization of continuous images of the space \(\omega_ 1+ 1\). Finally, the author proves a theorem independently proved by Bekkali. Theorem. A Boolean algebra with a non-Archimedean generating set is a subalgebra of an interval algebra. While the converse of this theorem is true in certain cases (e.g., the subalgebra is superatomic, a result due to Bonnet), the general case is not known. In particular, the counterexamples of this paper have non- Archimedean generating sets.
Mathematics Subject Classification ID: 06E05
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 54G12
zbMATH DE Number: 446216
zbMATH Keywords: Boolean algebra generated by left-closed cones
zbMATH Keywords: tree algebra
zbMATH Keywords: interval algebra on an ordinal
zbMATH Keywords: non-Archimedean families
zbMATH Keywords: clopen algebras
zbMATH Keywords: quotient spaces
reviewed by: Judith Roitman
MaRDI profile type: Publication
full work available at URL: 0166-8641(93)90149-8
OpenAlex ID: W2046744220
cites work: Q3994935
cites work: Pseudo-trees and Boolean algebras
cites work: Orderability of all noncompact images
description: scientific article; zbMATH DE number 446216