  It is known that there is no finitely presented group for which the Dehn
function lies asymptotically strictly between linear and quadratic functions.
This work presents an example of a semigroup that has Dehn function equivalent
to $n \log n$, thus it lies strictly inside the said gap. The example is
obtained by symmetrizing the rewriting rules of a particular semi-Thue system,
which has the derivational complexity function $n \log n$. We also show that
such connection is not universal by providing a semi-Thue system, for which the
Dehn function of the symmetrized semigroup asymptotically differs from the
derivational complexity of the initial system.
