The criterion of \(\tilde \alpha\)-monotonicity of Boolean functions (Q2730576)

Let be \(Z_2=\{0,1\}\), and let \(Z_2^{n}\) be the set of \(n\)-dimensional Boolean vectors. Each \(\widetilde\alpha\in Z_2^{n}\) generates the partial order relation \(\prec_{\widetilde\alpha}\) on the set \(Z_2^{n}\): \(\widetilde\beta \prec_{\widetilde\alpha}\widetilde\delta\) when for all \(i=1,2,\ldots,n\), \(\beta_{i}\leq\delta_{i}\) if \(\alpha_{i}=1\) and \(\beta_{i}\geq\delta_{i}\) if \(\alpha_{i}=0\). The Boolean function \(f: Z_2^{n}\to Z_2\) is called \(\widetilde \alpha\)-monotonic if, for arbitrary \(\widetilde\beta,\widetilde\delta\in Z_2^{n}\), from the condition \(\widetilde\beta \prec_{\widetilde\alpha}\widetilde\delta\) it follows that \(f(\widetilde\beta)\leq f(\widetilde\delta)\). The main result is the following. NEWLINENEWLINENEWLINEFor \(\widetilde \alpha\)-monotonicity of Boolean functions, and only for them, the \(\alpha\)-component of the \(K_{\widetilde \alpha}\)-energetic spectrum is equal to the norm of the Hamming function and all other components are equal to zero.