47 research outputs found
Filters, ideal independence and ideal Mr\'owka spaces
A family such that for all finite
and , the set is infinite, is
said to be ideal independent.
We prove that an ideal independent family is maximal if and
only if is -completely separable and maximal -almost disjoint for a particular ideal on . We show
that , where is the
minimal cardinality of maximal ideal independent family. This, in particular,
establishes the independence of and . Given
an arbitrary set of uncountable cardinals, we show how to simultaneously
adjoin via forcing maximal ideal independent families of cardinality
for each , thus establishing the consistency of . Assuming , we construct a maximal
ideal independent family, which remains maximal after forcing with any proper,
-bounding, -point preserving forcing notion and evaluate
in several well studied forcing extensions.
We also study natural filters associated with ideal independence and
introduce an analog of Mr\'owka spaces for ideal independent families.Comment: 18 Pages, subsumes arXiv:2206.1401
Infinite Permutation Groups
This is a transcript of a lecture course on Infinite Permutation Groups given
by Peter M. Neumann (1940-2020) in Oxford during the academic year 1988-1989.
The field of Infinite Permutation Groups only emerged as an independent field
of study in the 1980's. Most of the results described in these notes were at
the time of the lectures brand new and had either just recently appeared in
print or had not appeared formally. A large part of the results described is
either due to Peter himself or heavily influenced by him. These notes offer
Peter's personal take on a field that he was instrumental in creating and in
many cases ideas and questions that can not be found in the published
literature.Comment: The text of [v1] and [v2] is identical. [v1] is formatted so that
each new lecture starts at a right hand page (127+v pages) and [v2] is
formatted without all the extra page breaks (93+v pages). Edited by David A.
Craven (Birmingham), Dugald Macpherson (Leeds) and R\"ognvaldur G. M\"oller
(Reykjav\'ik). Comments and corrections can be sent to R\"oggi M\"oller
([email protected]
Definability of maximal cofinitary groups
We present a proof of a result, previously announced by the second author,
that there is a closed (even ) set generating an (even
) maximal cofinitary group (short, mcg) which is isomorphic to a
free group. In this isomorphism class, this is the lowest possible definitional
complexity of an mcg.Comment: This work is part of the first authors thesi
Higher Independence
We study higher analogues of the classical independence number on .
For regular uncountable, we denote by the minimal size of
a maximal -independent family. We establish ZFC relations between
and the standard higher analogues of some of the classical cardinal
characteristics, e.g. and
.
For measurable, assuming that we construct a
maximal -independent family which remains maximal after the
-support product of many copies of -Sacks forcing.
Thus, we show the consistency of
. We conclude the
paper with interesting open questions and discuss difficulties regarding other
natural approaches to higher independence
Cofinitary groups
Das Thema dieser Arbeit sind kofinitaÌre Gruppen, eine spezielle Klasse an Untergruppen der unendlichen Permutationsgruppen. Wir beginnen mit einer UÌbersicht der algebraischen Resultate fuÌr diese Gruppen. Die wichtigsten Resultate in diesem Kapitel sind strukturelle EinschraÌnkungen der KardinalitaÌt von kofinitaÌren Gruppen durch ihre Orbitstruktur. In weiterer Folge betrachten wir Konstruktionen von kofinitaÌren Gruppen mittels projektiver Limits und Automorphismen von Boolschen Algebren. Der Rest der Thesis befasst sich mit maximalen kofinitaÌren Gruppen, wobei wir zuerst die moÌglichen GroÌĂen, sowie die kombinatorische Charakteristik a_g betrachten. In Kapitel 4 werden wir Forcing verwenden, um zu jedem Tupel (n, m) in N>0 x N eine maximale kofinitaÌre Gruppe zu finden welche n unendliche und m endliche Orbits aufweist, wodurch wir unendlich viele nicht isomorphe Gruppen konstruieren koÌnnen. In Kapitel 5 konstruieren wir mittels Forcing eine maximale kofinitaÌre Gruppe in welche wir alle abzaÌhlbar unendlichen Gruppen einbetten koÌnnen. Im letzten Kapitel zeigen wir eine Konstruktion, welche uns die moÌglichen GroÌĂen von maximalen kofinitaÌren Gruppen in unserem Modell steuern laÌsst.The topic of this thesis is cofinitary groups, which are special subgroups of the infinite permutation group. We will begin by giving an overview of the algebraic properties of cofinitary groups. We will survey the algebraic properties of cofinitary groups, where the main results give us bounds on the size of cofinitary groups based on their orbit structure. We will then examine how to construct cofinitary groups using inverse limits and automorphisms of Boolean algebras. We then begin looking at maximal cofinitary groups and their possible sizes as well as the combinatorial characteristic a_g. In chapter 4 we will use forcing to show that there are infinitely many, non-isomorphic, maximal cofinitary groups, by constructing a group with n infinite and m finite orbits, for any tuple (n, m) in N>0 x N. In chapter 5, we use forcing constructions to show the existence of a maximal cofinitary group into which every countable group embeds. Finally, we show that we can tightly control the possible sizes of cofinitary groups in a model by adapting a novel proof from the theory of maximal almost disjoint families
MAXIMAL DISCRETE SETS (Set Theory and Infinity)
We survey results and open questions as well as give some new results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, L-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics ae, forE<w1 as well as the notions of spectra of characteristics and optimal projective witnesses. We give a streamlined account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding
Cardinal invariants of the continuum -A survey
Abstract These are expanded notes of a series of two lectures given at the meeting on axiomatic set theory at KyĆto University in November 2000. The lectures were intended to survey the state of the art of the theory of cardinal invariants of the continuum, and focused on the interplay between iterated forcing theory and cardinal invariants, as well as on important open problems. To round off the present written account of this survey, we also include sections on ZF C-inequalities between cardinal invariants, and on applications outside of set theory. However, due to the sheer size of the area, proofs had to be mostly left out. While being more comprehensive than the original talks, the personal flavor of the latter is preserved in the notes. Some of the material included was presented in talks at other conferences
Unsolved Problems in Group Theory. The Kourovka Notebook
This is a collection of open problems in group theory proposed by hundreds of
mathematicians from all over the world. It has been published every 2-4 years
in Novosibirsk since 1965. This is the 19th edition, which contains 111 new
problems and a number of comments on about 1000 problems from the previous
editions.Comment: A few new solutions and references have been added or update