Switching functions whose monotone complexity

A sequence of monotone switching functions fn:{0,1}n →{0,1}n is constructed, such that the monotone complexity of fn grows faster than O(n2-ε) for any ε>O. Previously the best lower bounds of this nature were several O(n3/2) bounds due to Pratt, Paterson, Mehlhorn/Galil and Savage. Then we discuss the complexity gap between monotone circuits and circuits over the basis {Λ,V,-} computing the functions, which we have examined before. We show, that for several sequences of monotone functions this gap is at least as large as the largest previously proved gap for sequences of explicitely defined monotone functions (the gap,which was proved for the Boolean matrix product).