From the authors’ abstract: Given an arbitrary Boolean algebra \(B\), every function \(f:B^{n}\rightarrow B\) that is Boolean (i.e., that has the canonical disjunctive form \(f(x)=\sum _{i}f(i)m_{i}(x)\)) can be expressed in the form \(f(x)=g(x,a)\), where \(g:B^{n+p}\rightarrow B\) is a simple Boolean function (i.e., \(g\) involves no constants from \(B\) or \(g=1\) or \(g=0\)) and \(a\in B^{p}\). The relationships between \(f\) and \(g\), and in particular between the equation \(f(x)=1\) and the equations \(g(x,j)=1\) (\(j=0,1,\ldots ,2^{p}-1\)), have been studied in the literature. In the present paper we follow this line of research. The main result provides three necessary and sufficient conditions for the equation \(g(x,a)=1\) to have a unique solution.