A semigroup with linearithmic Dehn function

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.