A semigroup with linearithmic Dehn function

It is known that no finitely presented group has the Dehn function asymptotically strictly between linear and quadratic functions. We present an example of a semigroup whose Dehn function is equivalent to \(n \log n\) $nlogn$ , thus 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 \(n \log n\) $nlogn$ .