A theory of representations for
compact semigroups has been lacking due in large part to the absence of a
translation-invariant carrying measure that exists for compact groups. The
object in this paper is to show that for a compact, group-extremal affine
semigroup there is a sufficient system of representations by linear operators on
finite-dimensional complex linear spaces; in the abelian case, a sufficient system of
affine semicharacters is obtained. As a result, a compact group-extremal affine
semigroup is the inverse limit of compact, finite-dimensional, group-extremal affine
semigroups.
