Template iterations and maximal cofinitary groups

Vera Fischer ¹   ; Asger Törnquist ²

¹ Institute of Discrete Mathematics and Geometry Technical University of Vienna Wiedner Hauptstrasse 8–10 1040 Wien, Austria
² Department of Mathematical Sciences University of Copenhagen Universitetspark 5 2100 København, Denmark

Fundamenta Mathematicae, Tome 230 (2015) no. 3, pp. 205-236

Résumé

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.

Détail

DOI : 10.4064/fm230-3-1

Keywords: brendle hechlers forcing notion adding maximal almost disjoint family along appropriate template forcing construction mathfrak minimal size maximal almost disjoint family countable cofinality main result present paper mathfrak minimal size maximal cofinitary group countable cofinality prove define natural poset adding maximal cofinitary group given cardinality which enjoys certain combinatorial properties allowing within similar template forcing construction additionally mathfrak minimal size maximal family almost disjoint permutations mathfrak minimal size maximal eventually different family countable cofinality