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Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers

Saharon Shelah, Juris Steprāns (2007)

Fundamenta Mathematicae

If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. The cardinal invariant A(G) is defined to be the least uncountable cardinal in the abelian subgroup spectrum of G. The value of A(G) is examined for various groups G which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the...

Potential isomorphism and semi-proper trees

Alex Hellsten, Tapani Hyttinen, Saharon Shelah (2002)

Fundamenta Mathematicae

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. ...

Products of small modules

Peter Kálnai, Jan Žemlička (2014)

Commentationes Mathematicae Universitatis Carolinae

Module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal.

Proper translation

Heike Mildenberger, Saharon Shelah (2011)

Fundamenta Mathematicae

We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that 2 ω = ℵ ₂ together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and “all Aronszajn trees are special” is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from [“Proper and Improper Forcing”, Ch. V]) and < ω₁-proper ω ω -bounding forcings adding reals. We show...

P-sets and minimal right ideals in ℕ*

W. R. Brian (2015)

Fundamenta Mathematicae

Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift...

Quotients of indecomposable Banach spaces of continuous functions

Rogério Augusto dos Santos Fajardo (2012)

Studia Mathematica

Assuming ⋄, we construct a connected compact topological space K such that for every closed L ⊂ K the Banach space C(L) has few operators, in the sense that every operator on C(L) is multiplication by a continuous function plus a weakly compact operator. In particular, C(K) is indecomposable and has continuum many non-isomorphic indecomposable quotients, and K does not contain a homeomorphic copy of βℕ. Moreover, assuming CH we construct a connected compact K where C(K) has few...

Ramseyan ultrafilters

Lorenz Halbeisen (2001)

Fundamenta Mathematicae

We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...

Reflecting Lindelöf and converging ω₁-sequences

Alan Dow, Klaas Pieter Hart (2014)

Fundamenta Mathematicae

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ω-sequence or a non-trivial converging ω₁-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ω₁-sequences is first-countable and, in addition,...

Regular spaces of small extent are ω-resolvable

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2015)

Fundamenta Mathematicae

We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies Δ(X) > e(X) is ω-resolvable. Here Δ(X), the dispersion character of X, is the smallest size of a non-empty open set in X, and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable. We also prove that any regular...

Resolvability in c.c.c. generic extensions

Lajos Soukup, Adrienne Stanley (2017)

Commentationes Mathematicae Universitatis Carolinae

Every crowded space X is ω -resolvable in the c.c.c. generic extension V Fn ( | X | , 2 ) of the ground model. We investigate what we can say about λ -resolvability in c.c.c. generic extensions for λ > ω . A topological space is monotonically ω 1 -resolvable if there is a function f : X → ω 1 such that { x ∈ X : f ( x ) ≥ α } ⊂ d e n s e X for each α < ω 1 . We show that given a T 1 space X the following statements are equivalent: (1) X is ω 1 -resolvable in some c.c.c. generic extension; (2) X is monotonically ω 1 -resolvable; (3) X is ω 1 -resolvable in the Cohen-generic extension V Fn ( ω 1 , 2 ) ....

Currently displaying 301 – 320 of 435