An old question of R. J. Koch is whether every compact semigroup embeds in a compact divisible semigroup. In this paper the authors employ a variety of techniques to extend the previously known positive results to various classes of compact commutative semigroups, called t-semigroups, in which every element lies on a semigroup multiplicatively isomorphic to [0,t] for some \(0\leq t<1\).