Boolean sets and most general solutions of Boolean equations
\textit{C. Reischer, D. A. Simovici, I. Stojmenovic} and \textit{R. Tosic} [Inf. Sci. 99, No.~3--4, 195--204 (1997; Zbl 0921.06012)] were the first to determine the internal structure of the set \({\mathcal C}_{f}\) of zeros of a Boolean function \(f\) in terms of ring sum, complementation and the difference \(x-y=xy'\). This paper gives a new construction of \({\mathcal C}_{f}\) and describes the union of the sets \({\mathcal C}_{f}\), called Boolean sets, when \(f\) runs over the set of evanescible Boolean functions of \(n\) variables (\(f\) is said to be evanescible if the Boolean equation \(f(X)=0\) is consistent, i.e., it has solutions \(X\in B^{n}\), where \(B\) is an arbitrary Boolean algebra). On this occasion the paper provides an intrinsic characterization of the most general subsumptive general solution and determines the systems of recurrent inequalities that are subsumptive general solutions of certain Boolean equations and the Boolean transformations that are parametric general solutions of certain Boolean equations. It is also shown that every Boolean transformation \(\Phi :B^{n}\rightarrow B^{n}\) is the parametric general solution of a certain Boolean equation. Four examples are considered.
scientific article; zbMATH DE number 5758540
scientific article