Let \(g:\Omega\longrightarrow\Omega'\) be a proper holomorphic mapping, where \(\Omega\subset\mathbb C^n\), \(\Omega'\subset\mathbb C^k\), \(k\geq n\). Given an algebra \(\mathcal B(U)\) of functions defined on a set \(U\subset\Omega\) (e.g., \(\mathcal P(K)\), \(\mathcal O(U)\), \(\mathcal A(U)\), \(\mathcal H^\infty(U)\)) the authors study properties of the subalgebra \[ \mathcal B_g(U):=\big\{f\in\mathcal B(U): \forall_{z,w\in U}:(g(z)=g(w) \Longrightarrow f(z)=f(w))\big\}. \] Under various additional assumptions, they investigate alternative representations of subalgebra \(\mathcal B_g(U)\) and characterize the fibers of their spectra. In particular, they study the algebras of functions invariant under permutations and the algebras of functions defined on the symmetrized polydisk.