  We present an example of a Thermal Process for a system of $d$ energy levels,
which cannot be performed without an instant access to the whole energy space.
This Thermal Process is uniquely connected with a transition between some
states of the system, that cannot be performed without access to the whole
energy space even when approximate transitions are allowed. Pursuing the
question about the decomposability of Thermal Processes into convex
combinations of compositions of processes acting non-trivially on smaller
subspaces, we investigate transitions within the subspace of states diagonal in
the energy basis. For three level systems, we determine the set of extremal
points of these operations, as well as the minimal set of operations needed to
perform an arbitrary Thermal Process, and connect the set of Thermal Processes
with thermomajorization criterion. We show that the structure of the set
depends on temperature, which is associated with the fact that Thermal
Processes cannot increase deterministically extractable work from a state --
the conclusion that holds for arbitrary $d$ level system. We also connect the
decomposability problem with detailed balance symmetry of an extremal Thermal
Processes.
