Vol. 34, No. 1, 1970

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Locally compact Clifford semigroups

James Wilson Stepp

Vol. 34 (1970), No. 1, 163–176

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.

Mathematical Subject Classification
Primary: 22.05
Received: 12 March 1969
Revised: 31 October 1969
Published: 1 July 1970
James Wilson Stepp