Two scaling functions \(\varphi _A\) and \(\varphi _B\) for Parseval frame wavelets are algebraically isomorphic, \(\varphi _A \simeq \varphi _B\), if they have matching solutions to their (reduced) isomorphic systems of equations. Let A and B be \(d\times d\) and \(s\times s\) expansive dyadic integral matrices with \(d, s\ge 1\) respectively and let \(\varphi _A\) be a scaling function associated with matrix A and generated by a finite solution. There always exists a scaling function \(\varphi _B\) associated with matrix B such that $$\begin{aligned} \varphi _B \simeq \varphi _A. \end{aligned}$$ An example shows that the assumption on the finiteness of the solutions can not be removed. An algebraic isomorphism with consistency has orthogonality as an invariant.