An HL -semigroup is defined to be a topological semigroup with the property that the Schützenberger group of each H \mathcal {H} -class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL -semigroups to separate points of S ; or equivalently, is S isomorphic to a strict projective limit of HL -semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL -semigroups to separate points of S .