  We analyse the concept of generalized Bose-Einstein condensation (g-BEC),
known since 1982 for the perfect Bose gas (PBG) in the Casimir (or anisotropic)
boxes. Our aim is to establish a relation between this phenomenon and two
concepts: the concept of long cycles and the Off-Diagonal-Long-Range-Order
(ODLRO), which are usually considered as some adequate way to describe the
standard BEC on the ground state for the cubic boxes. First we show that these
three criterions are equivalent in this latter case. Then, basing on a scaling
approach, we revise formu- lation of these concepts to prove that the
classification of the g-BEC into three types I,II,III, implies a hierarchy of
long cycles (depending on their size scale) as well as a hierarchy of ODLRO
which depends on the coherence length of the condensate.
