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Template iterations and maximal cofinitary groups

Volume 230 / 2015

Vera Fischer, Asger Törnquist Fundamenta Mathematicae 230 (2015), 205-236 MSC: Primary 03E17; Secondary 03E35. DOI: 10.4064/fm230-3-1

Abstract

Jörg Brendle (2003) used Hechler's forcing notion for adding a maximal almost disjoint family along an appropriate template forcing construction to show that $\mathfrak a$ (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that $\mathfrak a_g$, the minimal size of a maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial properties allowing it to be used within a similar template forcing construction. Additionally we find that $\mathfrak a_p$, the minimal size of a maximal family of almost disjoint permutations, and $\mathfrak a_e$, the minimal size of a maximal eventually different family, can be of countable cofinality.

Authors

  • Vera FischerInstitute of Discrete Mathematics
    and Geometry
    Technical University of Vienna
    Wiedner Hauptstrasse 8–10
    1040 Wien, Austria
    e-mail
  • Asger TörnquistDepartment of Mathematical Sciences
    University of Copenhagen
    Universitetspark 5
    2100 København, Denmark
    e-mail

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