The purpose of this paper is twofold. First, it is shown that the ideal structure of a semigroup with dense subgroup is closely related to its transformation group structure. -That is, if a left orbit through a given point is locally compact, then the members of this orbit are precisely those elements which generate the same left ideal as the given point. Secondly, the author gives a number of theorems which have as their goal the establishment of a natural product structure near a nonzero idempotent. Specifically the work of F. Knowles [111 is improved upon to include (1) the possibility of a nonconnected group; (2) the possibility of a nonsimply connected orbit; and (3) the case in which the boundary of the group is more than a single orbit. Introduction. Numerous papers ostensibly dealing with semigroups with identity on a manifold (and having their origins in [8]) have in reality dealt almost exclusively with the maximal group G and its closure. In fact, for many of the results so obtained, one may as well have assumed merely that one had a dense Lie subgroup. No exhaustive study has been made of this hypothesis, nor does this paper in any way attempt to summarize what has been done in this area. Results have been obtained, for example, concerning the question of when an arbitrary dense subgroup is open, and these results are totally ignored in the present paper. The emphasis in the present paper is twofold. First, it is shown that the transformation group structure inherent in the semigroup is intimately connected with its structure via Green's relations. Second, several results are obtained which make it possible to give a topological decomposition of the semigroup analogous to that obtained in [10]. Specifically, the following is a special case of Theorem 14 of the paper: Let S be a semigroup on a simply connected manifold. Suppose S has a completely simple kernel M and a dense connected Lie subgroup G. Let e be an idempotent in M Received by the editors April 2, 1973 and, in revised form, December 19, 1973 and July 22, 1974. AMS(MOS)subject classifications (1970). Primary 20M10, 22A15; Secondary 54H15, 57E99.