A closure property of regular languages (Q1113686)

Given a semigroup homomorphism \(\theta\) : \(X^*\to {\mathcal P}(Y^*)\) where \({\mathcal P}(Y^*)\) is the powerset of \(Y^*\), we show that the inverse image \(K\theta^{-1}\) of a regular set \(K\subseteq Y^*\) is regular, too. This theorem has several consequences generalizing previously known results.