  Several consistency notions are available for a lower prevision P assessed on
a set D of gambles (bounded random variables), ranging from the well known
coherence to convexity and to the recently introduced 2-coherence and
2-convexity. In all these instances, a procedure with remarkable features,
called (coherent, convex, 2-coherent or 2-convex) natural extension, is
available to extend P, preserving its consistency properties, to an arbitrary
superset of gambles. We analyse the 2-coherent and 2-convex natural extensions,
E_2 and E_2c respectively, showing that they may coincide with the other
extensions in certain, special but rather common, cases of `full' conditional
lower prevision or probability assessments. This does generally not happen if P
is a(n unconditional) lower probability on the powerset of a given partition
and is extended to the gambles defined on the same partition. In this framework
we determine alternative formulae for E_2 and E_2c. We also show that E_2c may
be nearly vacuous in some sense, while the Choquet integral extension is
2-coherent if P is, and bounds from above the 2-coherent natural extension.
Relationships between the finiteness of the various natural extensions and
conditions of avoiding sure loss or weaker are also pointed out.
