A deterministic incomplete automaton script A sign = Q, ∑, δ is partially monotonic if its state set Q admits a linear order such that each partial transformation δ(-, a) with a ∑ preserves the restriction of the order to the domain of the transformation. We show that if script A sign possesses an annihilator word w ∑*whose action is nowhere defined, then script A sign is annihilated by a word of length |Q| + {|Q|-1/2 and this bound is tight. © 2010 Allerton Press, Inc.