A fixed point characterization of cofinite languages (Q800497)

Let \({\mathcal F}_{\Sigma}\) denote the class of all functions \(f:\Sigma^*\to\Sigma^*\) such that for any denumerable language \(L\subset\Sigma^*\), f(L) and L are disjoint, where \(\Sigma^*\) is a free monoid generated by a finite, nonempty set \(\Sigma\). The author proves that the class of all cofinite languages over \(\Sigma\) is identical to the class of all Fix f for all \(f\in {\mathcal F}_{\Sigma}\), where Fix f\(=\{w\in\Sigma^*: f(w)=w\}\).