  We investigate the possibility of semigroup extensions of the isometry group
of an identification space, in particular, of a compactified spacetime arising
from an identification map $p: \RR^n_t \to \RR^n_t / \Gamma$, where $\RR^n_t$
is a flat pseudo-Euclidean covering space and $\Gamma$ is a discrete group of
primitive lattice translations on this space. We show that the conditions under
which such an extension is possible are related to the index of the metric on
the subvector space spanned by the lattice vectors: If this restricted metric
is Euclidean, no extensions are possible. Furthermore, we provide an explicit
example of a semigroup extension of the isometry group of the identification
space obtained by compactifying a Lorentzian spacetime over a lattice which
contains a lightlike basis vector. The extension of the isometry group is shown
to be isomorphic to the semigroup $(\ZZ^{\times},\cdot)$, i.e. the set of
nonzero integers with multiplication as composition and 1 as unit element. A
theorem is proven which illustrates that such an extension is obstructed
whenever the metric on the covering spacetime is Euclidean.
