In this paper,
S
S
is an abelian semigroup on an
n
{\text {n}}
-dimensional simply connected manifold with boundary whose interior is a dense, simply connected, connected Lie group. We also assume there is a vector semigroup
V
k
−
V_k^ -
in
S
S
such that the interior of
S
S
misses the boundary of
V
k
−
V_k^ -
, and such that
(
S
−
G
L
k
)
/
V
k
(S - G{L_k})/{V_k}
is a group. It is shown that if
k
=
n
k = n
, then
S
S
is iseomorphic to
V
n
−
V_n^ -
, and if
k
=
1
,
2
k = 1,2
, or
n
−
1
n - 1
, then
S
S
is iseomorphic to
V
n
−
k
×
V
k
−
{V_{n - k}} \times V_k^ -
.