  We consider the Hamiltonian renormalisation group flow of discretised
one-dimensional physical theories. In particular, we investigate the influence
the choice of different embedding maps has on the RG flow and the resulting
continuum limit, and show in which sense they are, and in which sense they are
not equivalent as physical theories. We are furthermore elucidating the
interplay of the RG flow and the algebras operators satisfy, both on the
discrete and the continuum. Further, we propose preferred renormalisation
prescriptions for operator algebras guaranteeing to arrive at preferred
algebraic relations in the continuum, if suitable extension properties are
assumed. Finally, we introduce a weaker form of distributional equivalence, and
show how unitarily inequivalent continuum limits, which arise due to a choice
of different embedding maps, can still be weakly equivalent in that sense.
