, Volume 30, Issue 1, pp 1-11

Groupwise density and related cardinals

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Abstract

We prove several theorems about the cardinal \(\mathfrak{g}\) associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps fromω toω, all families of size \(< \mathfrak{g}\) are below all unbounded families. With respect to a natural ordering of filters onω, all filters generated by \(< \mathfrak{g}\) sets are below all non-feeble filters. If \(\mathfrak{u}< \mathfrak{g}\) then \(\mathfrak{b}< \mathfrak{u}\) and \(\mathfrak{g} = \mathfrak{d} = \mathfrak{c}\) . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from \(\mathfrak{u}< \mathfrak{g}\) by Laflamme are shown to be equivalent to \(\mathfrak{u}< \mathfrak{g}\) .