Three notions of a basis for an
ultrafilter in a Boolean algebra are investigated in this paper, namely having an
independent set of generators, a weakly independent set of generators and a weakly
independent set of generators over a proper subfilter. In general these three notions
are distinct, but for a Boolean algebra with an ordered base the latter two are
equivalent. This paper shows that a large class of Boolean algebras do not possess
ultrafilters with a basis, in particular no infinite homomorphic image of a σ-complete
Boolean algebra has a nonprincipal ultrafilter with a basis. For Boolean
algebras with an ordered base necessary and sufficient conditions on the
order type of the base are given for the Boolean algebra to have the basis
property.
