  Motivated by current searches for signals of Lorentz symmetry violation in
nature and recent investigations on generalized uncertainty principle (GUP)
models in anisotropic space, in this paper we identify GUP models satisfying
two criteria: (i) invariance of commutators under canonical transformations,
and (ii) physical independence of position and momentum on the ordering of
auxiliary operators in their definitions. Compliance of these criteria is
fundamental if one wishes to unambiguously describe GUP using an algebraic
approach but, surprisingly, neither is trivially satisfied when GUP is assumed
within anisotropic space. As a consequence, we use these criteria to place
important restrictions on what or how GUP models may be approached
algebraically.
