Robert Bonnet

On well-generated Boolean algebras

Abstract A Boolean algebra B that has a well-founded sublattice L which generates B is called a well-generated (WG) Boolean algebra. If in addition, L is generated by a complete set of representatives for B (see Definition 1.1), then B is said to be canonically well-generated (CWG). Every WG Boolean algebra is superatomic. We construct two basic examples of superatomic non well-generated Boolean algebras. Their cardinal sequences are 〈ℵ 1 ,ℵ 0 ,ℵ 1 ,1〉 and 〈ℵ 0 ,ℵ 0 ,2 ℵ 0 ,1〉 . Assuming MA ∧ (ℵ 1 ℵ 0 ) , we show that every algebra with one of the cardinal sequences 〈ℵ 0 : i 〈λ,ℵ 1 ,1〉 , α 1 , λ ℵ 0 , or 〈ℵ 0 ,2 ℵ 0 ,ℵ 1 ,1〉 is CWG. Assuming CH, or alternatively assuming MA ∧ (2 ℵ 0 =ℵ 2 ) , we determine which cardinal sequences admit only WG Boolean algebras. We find a necessary and sufficient condition for the canonical well-generatedness of algebras whose cardinal sequence has the form 〈ℵ 0 : i 〈λ,1〉 , α 1 . We conclude that if such an algebra is CWG, then all of its quotients are CWG. We show that the above is not true for general Boolean algebras. We also conclude that if the cardinality of such an algebra is less than the cardinal b defined below, then it is CWG. The cardinal b is the least cardinality of an unbounded subset of {f | f : ω → ω} . We investigate questions concerning embeddability, quotients and subalgebras of WG and CWG Boolean algebras, and construct various counter-examples.

ON POSET BOOLEAN ALGEBRAS

Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {xp : p∈P}, and the set of relations is {xp⋅xq=xp : p≤q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤B|G) is well-founded. A well-generated algebra is superatomic.THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated.The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements.THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B.This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra. Remember that an interval algebra is a special case of a poset algebra.

Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra

Let us recall that a Boolean algebra is superatomic if every subalgebra is atomic. So by the definition, every subalgebra of a superatomic algebra is superatomic. An obvious example of a superatomic algebra is the interval algebra generated by a well-ordered chain. In this work, we show that every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra, that is by definition, an interval algebra generated by a well-ordered chain. As a corollary, if B is an infinite superatomic subalgebra of an interval algebra, then B and the set At(B) of atoms of B have the same cardinality

On HCO spaces. An uncountable compactT 2 space, different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces, which is homeomorphic to each of its uncountable closed subspaces

AbstractLetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):(a)Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.(b)Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.Theorem (S. Shelah):Assume

A scattering of orders

\diamondsuit _{\aleph _1 }

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

. Then there is a HCO compact space X of Cantor-Bendixson rankω1} and of cardinality ℵ1 such that:(1)X has only countably many isolated points,(2)Every closed subset of X is countable or co-countable,(3)Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and(4)X is retractive. In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.

On poset Boolean algebras of scattered posets with finite width

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L*, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ω·2=ω+ω and 2·ω=ω). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals κ, λ≥0, let L(κ, λ)=κ + 1 + λ* . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form α + 1 + Σi<nL(κi, λi ), where α is any ordinal, n∈ω, for every i<n, κi, λi are regular cardinals and κi⩾λi, and if n>0, then α⩾max({κi: i<n}) · ω. This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).

Means on scattered compacta

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class