Remarks on affine semigroups

Abstract

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. It 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].It 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, H u , 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 P(A) = {x I x G S and xA = A}.As is customary, AB denotes the set of all products ab with aÇ^A and &G-Z? and we generally write x in place of \x\. It will be convenient to abbreviate P(S) by P. The structure of P is known in the following sense-supposing that PT^D the set PP\E^D and is indeed the set of left units of 5, E being the set of idempotents. Moreover, if e^.PC\E then Pe is a maximal subgroup of S and the assignment (#, y)->xy is an iseomorphism (topological isomorphism) of PeX(Pr\E) onto P. The following is a corollary to the principal result of [4]: THEOREM 1. If S is compact and convex and if ST^K then P(F) = P(S) C F.