Groupwise density and related cardinals

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$\mathfrak{u}< \mathfrak{g}$.