  We analyze the extension of the GUP theory deriving from the modified
uncertainty principle in agreement with the string low energy limit, which
represents one of the most general formulations satisfying the Jacobi identity,
in the context of the associative algebras. After providing some physical
insights on the nature of the considered approaches exploiting the cosmological
arena, first, we show how a natural formulation of the theory in an infinite
momentum space does not lead to the emergence of a nonzero minimal uncertainty
in position, then we construct a truncated formulation of the theory in
momentum space, proving that only in this case we can recover the desired
feature of the presence of a nonzero minimal uncertainty in position, which -
as usual in these theories - can be interpreted as a phenomenological
manifestation of cut-off physics effects. Both quantization schemes are
completely characterized and finally applied to study wave packets' behavior
and their evolution in time. The obtained results can shed light on which
generalizations of the GUP theory are more coherent with the string low energy
limit, in view of the existence of a minimum length in the form of a minimal
uncertainty in position.
