Small tree algebras with nontree subalgebras
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.
Boolean algebra generated by left-closed cones
scientific article; zbMATH DE number 446216