Remarks on affine semigroups
Heron Sherwood Collins · Pacific Journal of Mathematics · 1962
A problem of fundamental importance in the study of measure semigroups is the following: if S is a compact topological semigroup and S is the convolution semigroup (with the weak-* topology) of nonnegative normalized regular Borel measures on S, what relationship exists between a measure μ in S and its carrier?In the paper numbered [9, Lemma 5], Wendel proved that when S is a group and μ an idempotent in S, then carrier μ is a group and μ is Haar measure on carrier μ.He proved further that the mapping μ -•> carrier μ is a one-to-one mapping from the set of idempotents of S onto the set of closed subgroups of S. Glicksberg in [6] extended these results to the case when S is an abelian semigroup, In addition he showed (when S is a group or an abelian semigroup) the structure of the closed subgroups of S to be quite simple: each closed subgroup of S consists of the G-translates of Haar measure on some closed normal subgroup of a suitably^ chosen closed group G of S.It is our purpose in this paper to prove in § 2 that these properties are equivalent in general, each being equivalent to several other properties of some interest (see Theorem 2).One of these conditions is the geometric requirement that S can contain no 'parallelogram' whose vertices are μ, v 9 μv, and vμ, with all four of these measures idempotent and μ and v distinct.A crucial lemma of independent interest is that found in Theorem 1 of § 1, where it is shown that a line segment of an aίϊine semigroup (see [3] for definitions) which contains three distinct idempotents consists entirely of idempotents.Several corollaries are drawn from this theorem, among them the result that a compact affine semigroup consists of idempotents (i.e., is a band in the sense of [2]) if and only if it is rectangular, and that this occurs if and only if it is simple (i.e., contains no proper ideals).References for terminology and notation used here may be found in [3,6,8,9].l General affine semigroups.This section is devoted primarily to several results about general affine semigroups.However, we list first without proof two lemmas ([3, Theorem 3] and [4, Theorem 2]) needed in the sequel.LEMMA 1.Let T be a compact a fine topological semigroup and