Let S be a locally compact
Hausdorff semigroup which is a disjoint union of subgroups one of which is dense. If
S the disjoint union of exactly two groups one of which is compact, then S has been
completely described by K. II. Hofmann, and if S is the disjoint union of two
subgroups where the dense subgroup G has the added property that it is abelian and
G∕G0 is a union of compact groups, then S has been described in a previous paper of
the author.
It is the purpose of this paper to consider S when each subgroup of S is a
topological group when given the relative topology and G (the dense subgroup)
has the added property that it is abelian and G∕G0 is a union of compact
groups. In particular, we show how to reduce such a semigroup to a semigroup
which is a union of real vector groups (§3). In §4 we give the structure of S
under the added assumption that E(S) is isomorphic to E[(Rx)n], where
(Rx)n denotes the n-fold product of the nonnegative real numbers under
multiplication.
