On generalized language equations

A system of generalized language equations over an alphabet A is a set of n equations in n variables: X_i = G_i(X₁,..., X_n), i = 1,...,n, where the G_i are functions from P(A^*)]ⁿ into P(A^*), i=1,..., n, P(A^*) denoting the set of all languages over A. Furthermore the G_i are expressible in terms of set-operations, concatenations, and stars which involve the variable X_i as well as certain mixed languages. In this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations. Furthermore we show that a solution is regular if all fixed languages are regular.