Remarks on Affine Semigroups

A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication, denoted by juxtaposition. In what follows S will denote one such and it will be assumed that S is compact. I t thus entails no loss of generality to suppose that S is contained in a locally convex linear topological space 9C, but no particular imbedding is assumed. For general notions about semigroups we refer to [3] and for information concerning linear spaces to [2]. I t has been known for some time [3] that if 9C is finite dimensional, if S is convex (recall that S is compact) and if S has a unit (always denoted by u) then the maximal subgroup, Hu, which contains u is a subset of the boundary of S relative to 9C. Let F denote the boundary of 5, K the minimal ideal of S and, for any subset A of 5, let