47 research outputs found

    Filters, ideal independence and ideal Mr\'owka spaces

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    A family A⊆[ω]ω\mathcal{A} \subseteq [\omega]^\omega such that for all finite {Xi}i∈n⊆A\{X_i\}_{i\in n}\subseteq \mathcal A and A∈A∖{Xi}i∈nA \in \mathcal{A} \setminus \{X_i\}_{i\in n}, the set A∖⋃i∈nXiA \setminus \bigcup_{i \in n} X_i is infinite, is said to be ideal independent. We prove that an ideal independent family A\mathcal{A} is maximal if and only if A\mathcal A is J\mathcal J-completely separable and maximal J\mathcal J-almost disjoint for a particular ideal J\mathcal J on ω\omega. We show that u≀smm\mathfrak{u}\leq\mathfrak{s}_{mm}, where smm\mathfrak{s}_{mm} is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of smm\mathfrak{s}_{mm} and i\mathfrak{i}. Given an arbitrary set CC of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality λ\lambda for each λ∈C\lambda\in C, thus establishing the consistency of C⊆spec(smm)C\subseteq \hbox{spec}(\mathfrak{s}_{mm}). Assuming CH\mathsf{CH}, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, ωω^\omega\omega-bounding, pp-point preserving forcing notion and evaluate smm\mathfrak{s}_{mm} 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

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    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

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    We present a proof of a result, previously announced by the second author, that there is a closed (even Π10\Pi^0_1) set generating an FσF_\sigma (even ÎŁ20\Sigma^0_2) 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

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    We study higher analogues of the classical independence number on ω\omega. For Îș\kappa regular uncountable, we denote by i(Îș)i(\kappa) the minimal size of a maximal Îș\kappa-independent family. We establish ZFC relations between i(Îș)i(\kappa) and the standard higher analogues of some of the classical cardinal characteristics, e.g. r(Îș)≀i(Îș)\mathfrak{r}(\kappa)\leq\mathfrak{i}(\kappa) and d(Îș)≀i(Îș)\mathfrak{d}(\kappa)\leq\mathfrak{i}(\kappa). For Îș\kappa measurable, assuming that 2Îș=Îș+2^\kappa=\kappa^+ we construct a maximal Îș\kappa-independent family which remains maximal after the Îș\kappa-support product of λ\lambda many copies of Îș\kappa-Sacks forcing. Thus, we show the consistency of Îș+=d(Îș)=i(Îș)<2Îș\kappa^+=\mathfrak{d}(\kappa)=\mathfrak{i}(\kappa)<2^\kappa. We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence

    Cofinitary groups

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    Das Thema dieser Arbeit sind kofinitäre Gruppen, eine spezielle Klasse an Untergruppen der unendlichen Permutationsgruppen. Wir beginnen mit einer Übersicht der algebraischen Resultate für diese Gruppen. Die wichtigsten Resultate in diesem Kapitel sind strukturelle Einschränkungen der Kardinalität von kofinitären Gruppen durch ihre Orbitstruktur. In weiterer Folge betrachten wir Konstruktionen von kofinitären Gruppen mittels projektiver Limits und Automorphismen von Boolschen Algebren. Der Rest der Thesis befasst sich mit maximalen kofinitären Gruppen, wobei wir zuerst die möglichen GroÌˆĂŸen, 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 kofinitäre Gruppe zu finden welche n unendliche und m endliche Orbits aufweist, wodurch wir unendlich viele nicht isomorphe Gruppen konstruieren können. In Kapitel 5 konstruieren wir mittels Forcing eine maximale kofinitäre Gruppe in welche wir alle abzählbar unendlichen Gruppen einbetten können. Im letzten Kapitel zeigen wir eine Konstruktion, welche uns die möglichen GroÌˆĂŸen von maximalen kofinitären Gruppen in unserem Modell steuern lässt.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)

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    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

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    Abstract These are expanded notes of a series of two lectures given at the meeting on axiomatic set theory at Kyƍto 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

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    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
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