  We study the classical and quantum oscillator in the context of a
non-additive (deformed) displacement operator, associated with a
position-dependent effective mass, by means of the supersymmetric formalism.
From the supersymmetric partner Hamiltonians and the shape invariance technique
we obtain the eigenstates and the eigenvalues along with the ladders operators,
thus showing a preservation of the supersymmetric structure in terms of the
deformed counterpartners. The deformed space in supersymmetry allows to
characterize position-dependent effective mass, uniform field interactions and
to obtain a generalized uncertainty relation (GUP) that behaves as a
distinguishability measure for the coherent states, these latter satisfying a
periodic evolution of the GUP corrections.
