Reflectiveness and compression of threshold transformations (Q1841890)

A Boolean \(n\)-tuple is a function from the residue class ring with \(n\) elements in the Boolean algebra with two elements. A Boolean function (of \(n\) arguments) is a function from the set of all Boolean \(n\)-tuples in the Boolean algebra with two elements. A Boolean transformation is a transformation of a Boolean \(n\)-tuple. A Boolean isometry is a finite product of permutations and complementations of specific coordinates of some Boolean \(n\)-tuple. The Boolean isometries form a transformation group with \(n!2^n\) elements. Let \(T\) be a Boolean isometry. A one-one transformation \(F\) is reflective through \(T\) if \(F^{-1}=T^{-1}FT\). The paper proves that any Boolean isometry is reflective through some Boolean isometry of order \(2\). Particular cases studied concern so-called threshold transformations and self-dual transformations.