Monotone Versus Positive

A number of famous theorems about first-order logic were disproved in [60] in the case of finite structures, but Lyndon’s theorem on monotone vs. positive resisted the attack. It is defeated here. The counter-example gives a uniform sequence of constant-depth polynomial-size (functionally) monotone boolean circuits not equivalent to any (however nonuniform) sequence of constant-depth polynomial-size positive boolean circuits.