Transversally bounded lattices

Abstract

A problem stemming from a boundedness question for torsion modules and its translation into ideal lattices is explored in the setting of abstract lattices. Call a complete lattice L transversally bounded (resp., uniformly transversally bounded) if for all families (X _i)_i∈Iof nonempty subsets of L with the property that ∨{x _i‖i∈I}<1 for all choices of x _i∈X _i, almost all of the sets X _ihave join smaller than 1 (resp., ∪ _j∈J X _jhas join smaller than 1 for some cofinite subset J of I). It is shown that the lattices which are transversally bounded, but not uniformly so, correspond to certain ultrafilters with peculiar boundedness properties similar to those studied by Ramsey. The prototypical candidates of the two types of lattices which one is led to construct from ultrafilters (in particular the lattices arising from what will be called ‘Ramsey systems’) appear to be of interest beyond the questions at stake.