200 research outputs found
Partial orderings with the weak Freese-Nation property
A partial ordering P is said to have the weak Freese-Nation property (WFN) if
there is a mapping f:P ---> [P]^{<= aleph_0} such that, for any a, b in P, if a
<= b then there exists c in f(a) cap f(b) such that a <= c <= b. In this note,
we study the WFN and some of its generalizations. Some features of the class of
BAs with the WFN seem to be quite sensitive to additional axioms of set theory:
e.g., under CH, every ccc cBA has this property while, under b >= aleph_2,
there exists no cBA with the WFN
A game on partial orderings
We study the determinacy of the game G_kappa (A) introduced in [FKSh:549] for
uncountable regular kappa and several classes of partial orderings A. Among
trees or Boolean algebras, we can always find an A such that G_kappa (A) is
undetermined. For the class of linear orders, the existence of such A depends
on the size of kappa^{< kappa}. In particular we obtain a characterization of
kappa^{< kappa}= kappa in terms of determinacy of the game G_kappa (L) for
linear orders L
Complements and quasicomplements in the lattice of subalgebras of P(ω)
AbstractIn the lattice of subalgebras of a Boolean algebra D call Ba 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 Martin's Axiom, describe the isomorphism types of some quasicomplements of the finite-cofinite subalgebra of P(ω)
Covering dimension and finite-to-one maps
Hurewicz' characterized the dimension of separable metrizable spaces by means
of finite-to-one maps. We investigate whether this characterization also holds
in the class of compact F-spaces of weight c. Our main result is that, assuming
the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the
continuous image of a zero-dimensional compact Hausdorff space by an at most
2n-to-1 map
On Property (FA) for wreath products
We characterize permutational wreath products with Property (FA). For
instance, the standard wreath product A wr B of two nontrivial countable groups
A,B, has Property (FA) if and only if B has Property (FA) and A is a finitely
generated group with finite abelianisation. We also prove an analogous result
for hereditary Property (FA). On the other hand, we prove that many wreath
products with hereditary Property (FA) are not quotients of finitely presented
groups with the same property.Comment: 12 pages, 0 figur
Strongly bounded groups and infinite powers of finite groups
We define a group as strongly bounded if every isometric action on a metric
space has bounded orbits. This latter property is equivalent to the so-called
uncountable strong cofinality, recently introduced by G. Bergman.
Our main result is that G^I is strongly bounded when G is a finite, perfect
group and I is any set. This strengthens a result of Koppelberg and Tits. We
also prove that omega_1-existentially closed groups are strongly bounded.Comment: 10 pages, no figure. Versions 1-3 were entitled "Uncountable groups
with Property (FH)". To appear in Comm. Algebr
Sequential closure in the space of measures
We show that there is a compact topological space carrying a measure which is
not a weak* limit of finitely supported measures but is in the sequential
closure of the set of such measures. We construct compact spaces with measures
of arbitrarily high levels of complexity in this sequential hierarchy. It
follows that there is a compact space in which the sequential closure cannot be
obtained in countably many steps. However, we show that this is not the case
for our spaces where the sequential closure is always obtained in countably
many steps.Comment: (18 pages, a gap in an argument from the previous version fixed
On Efimov spaces and Radon measures
We give a construction under CH of an infinite Hausdorff compact space having no converging sequences and carrying no Radon measure of uncountable type. Under ? we obtain another example of a compact space with no convergent sequences, which in addition has the stronger property that every nonatomic Radon measure on it is uniformly regular. This example refutes a conjecture of Mercourakis from 1996 stating that if every measure on a compact space K is uniformly regular then K is necessarily sequentially compact
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