  We derive generalised uncertainty relations (GURs) for angular momentum and
spin in the smeared-space model of quantum geometry. The model implements a
minimum length and a minimum linear momentum, and recovers both the generalised
uncertainty principle (GUP) and the extended uncertainty principle (EUP) within
a single formalism. In this paper, we investigate the consequences of these
results for particles with extrinsic and intrinsic angular momentum, and obtain
generalisations of the canonical ${\rm so(3)}$ and ${\rm su(2)}$ algebras. We
find that, although ${\rm SO(3)}$ symmetry is preserved on three-dimensional
slices of an enlarged phase space, individual subcomponents of the generalised
generators obey nontrivial subalgebras. These give rise to GURs for angular
momentum while leaving the canonical commutation relations intact except for a
simple rescaling, $\hbar \rightarrow \hbar + \beta$. The value of the new
parameter, $\beta \simeq \hbar \times 10^{-61}$, is determined by the ratio of
the dark energy density to the Planck density. Here, we assume the former to be
of the order of the Planck length and the latter to be of the order of the de
Sitter momentum $\sim \hbar\sqrt{\Lambda}$, where $\Lambda$ is the cosmological
constant. In the smeared-space model, $\hbar$ and $\beta$ are interpreted as
the quantisation scales for matter and geometry, respectively, and a quantum
state vector is associated with the spatial background. We show that this also
gives rise to a rescaled Lie algebra for generalised spin operators, together
with associated subalgebras that are analogous to those for orbital angular
momentum. Remarkably, consistency of the algebraic structure requires the
quantum state associated with a flat background to be fermionic, with spin
eigenvalues $\pm \beta/2$. Finally, the modified spin algebra leads to GURs for
spin measurements.
