Sabine Koppelberg

Boolean algebras as unions of chains of subalgebras

where, for a < h, B,~ is a subalgebra of B and a </3 < h implies B~ _ B~, B~ ~: B~ (we identify cardinals with initial ordinals and each ordinal with the set of smaller ordinals). The question how cf(B) does look like makes sense, of course, for other types of structures than Boolean algebras; see [2] and [5]. It is proved in [2] that there is a group G s.t. cf(G) is uncountable, one of the arguments being that there is a BA B s.t. cf(B) is uncountable. We shall show that in all cases we are able to handle, cf(B) equals No or N1, but we do not know whether this holds for every BA B. We shall denote the finitary operations on a BA B by + , . , , 0, 1; the infinite ones by Y. and l-[. Let us first remark that of(B) is well-defined: suppose card(B)=h-->No. Let B = {b~ I a < )t} and define, for a < h, B~ to be the subalgebra of B generated by {b~[~< a}. Each B~ is a proper subalgebra of B (for h = No, this holds since a finitely generated BA is finite), and thus, cf(B)-< h = card(B). There are some very simple facts:

Partial orderings with the weak Freese-Nation property

Abstract A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping tf : P → [P] ⩽ℵ 0 such that, for any a, b ϵ P, if a ⩽ b then there exists c ϵ tf(a)∩tf(b) such that a ⩽ c ⩽ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, every ccc complete Boolean algebra has this property while, under b ⩾ ℵ 2 , there exists no complete Boolean algebra with the WFN (Theorem 6.2).

Maximal chains in interval algebras

A maximal chain in a Boolean algebra A is a subset of A which is maximal with respect to the property of being totally ordered under the Boolean (partial) order of A. And A is generated by some chain C ~_ A iff A is isomorphic to an interval algebra (cf. Section 1 for the relevant definitions). More precisely, if A is the interval algebra Intalg X of a linear order X, then A is generated by some maximal chain C ~_ A isomorphic (up to addition or deletion of endpoints) to X. Thus it seems to be a natural question which interval algebras have the property that all of their maximal chains are isomorphic. E.g.: are all maximal chains of the interval algebra of the reals isomorphic? This is easily answered by part of the following observation.

Homogeneous boolean algebras may have non-simple automorphism groups

Abstract We construct, under CH, a homogeneous Boolean algebra A such that A has a countable dense subalgebra and cardinality ω 1 and the automorphism group of A is not simple. Under MA+ω 1 ω , the automorphism group of such an algebra is simple. Moreover, we prove that the automorphism group of any infinite free Boolean algebra is simple.

On L∞κ-free Boolean algebras

Abstract We study L ∞κ -freeness in the variety of Boolean algebras. It is shown that some of the theorems on L ∞κ -free algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean algebras. But we also investigate properties such as the ccc of L ∞κ -free Boolean algebras which have no counterpart in the varieties above.

A game on partial orderings

Abstract We study the determinacy of the game G κ ( A ) introduced in Fuchino, Koppelberg and Shelah (to appear) for uncountable regular κ and several classes of partial orderings A . Among trees or Boolean algebras, we can always find an A such that G κ ( A ) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ . In particular we obtain a characterization of κ = κ in terms of determinacy of the game G κ ( L ) for linear orders L .

DENSITIES OF ULTRAPRODUCTS OF BOOLEAN ALGEBRAS 1

We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density …A resp. the topological density dA of a Boolean algebra A commutes with formation of ul- traproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.

Complements and quasicomplements in the lattice of subalgebras of P(ω)

Abstract In the lattice of subalgebras of a Boolean algebra D call B a complement of A if A ∩ B = {0,1} and { A ∪ B } generates D. B is called a quasicomplement of A if it is maximal w.r.t. the property A ∩ B = {0, 1}. We characterize those countable subalgebras of P ( ω ) which have a complement, and, assuming Martins Axiom, describe the isomorphism types of some quasicomplements of the finite-cofinite subalgebra of P ( ω ).

Groups of permutations with few fixed points

Let n be a non-zero cardinal (finite or infinite). A group G of permutat ions of a set X is n-transitive if (a) for every two injective n-tuples a, b in X, there is at least one g ~ G such that g ( a ) = b (i.e. g(a~)=b~ for i<n). G is said to be sharply n-transitive if G satisfies (a) plus (b) for every two injective n-tuples a, b in X, there is at most one g ~ G such that g(a) = b. For a permutat ion g of X, let

A superatomic Boolean algebra with few automorphisms

Abstract. Assuming GCH, we prove that for every successor cardinal μ > ω1, there is a superatomic Boolean algebra B such that |B| = 2μ and |Aut B| = μ. Under ◊ω1, the same holds for μ = ω1. This answers Monks Question 80 in [Mo].