On symmetric boolean functions invariant under the Möbius transform

The work is devoted to the study of the class of Boolean functions that are invariant under the Möbius transform. In the first part of the paper, we systematize general information on the Möbius transform and its fixed points. In the second part, we consider a class of symmetric Boolean functions that are invariant under the Möbius transform. The relationship of these functions with columns of the Sierpinski triangle is shown. We propose a method for obtaining masks of all such functions as sums of columns of the Sierpinski triangle. For the case $n=2^m-1$, we proved that a symmetric function is invariant if and only if its mask is invariant.