  We construct two types of scalar field theory on Snyder space-time. The first
one is based on the natural momenta addition inherent to the coset momentum
space. This construction uncovers a non-associative deformation of the
Poincar\'e symmetries. The second one considers Snyder space-time as a subspace
of a larger non-commutative space. We discuss different possibilities to
restrict the extra-dimensional scalar field theory to a theory living only on
Sndyer space-time and present the consequences of these restrictions on the
Poincar\'e symmetries. We show moreover how the non-associative approach and
the Doplicher-Fredenhagen-Roberts space can be seen as specific approximations
of the extra-dimensional theory. These results are obtained for the 3d
Euclidian Snyder space-time constructed from $\SO(3,1)/\SO(3)$, but our results
extend to any dimension and signature.
