  Fell's absorption principle states that the left regular representation of a
group absorbs any unitary representation of the group when tensored with it. In
a weakened form, this result carries over to the left regular representation of
a right LCM submonoid of a group and its Nica covariant isometric
representations but it fails if the semigroup does not satisfy independence. In
this paper we explain how to extend Fell's absorption principle to an arbitrary
submonoid $P$ of a group $G$ by using an enhanced version of the left regular
representation. Li's semigroup $C^*$-algebra $C^*_s(P)$ and its representations
appear naturally in our context. Using the enhanced left regular
representation, we not only provide a very concrete presentation for the
reduced object for $C^*_s(P)$ but we also obtain very transparent proofs of
earlier results. We also address the non-selfadjoint theory and we show that
the non-selfadjoint object attached to the enhanced left regular representation
coincides with that of the left regular representation. We obtain a
non-selfadjoint version of Fell's absorption principle involving the tensor
algebra of a semigroup and we use it to improve recent results of Clouatre and
Dor-On on the residual finite dimensionality of certain $C^*$-algebras
associated with such tensor algebras. We also give yet another proof for the
existence of a $C^*$-algebra which is co-universal for equivariant,
Li-covariant representations of a submonoid $P$ of a group $G$.
