Linear transformations of monotone functions on the discrete cube

For a function f : {0. 1}(n) -> R and an invertible linear transformation L is an element of GL(n)(2), we consider the function Lf : {0, 1}(n) -> R defined by Lf (x) = f (Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf) >= I(f), where I(f) is the total influence off. Second. we conjecture that if both f and L(f) are monotone, then f = L(f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular. (C) 2009 Elsevier B.V. All rights reserved.