  Partially greedy bases in Banach spaces were introduced by Dilworth et al. as
a strictly weaker notion than the (almost) greedy bases. In this paper, we
study two natural ways to strengthen the definition of partial greediness. The
first way produces what we call the consecutive almost greedy property, which
turns out to be equivalent to the almost greedy property. Meanwhile, the second
way reproduces the PG property for Schauder bases but a strictly stronger
property for general bases.
