We consider the fragment Sigma^2_m of two-variable first-order logic FO^2[<] over finite words which is defined by restricting the nesting depth of negations to at most m. Our first result is a combinatorial characterization of Sigma^2_m in terms of so-called rankers. This generalizes a result by Weis and Immerman which we recover as an immediate consequence.

Our second result is an effective algebraic characterization of Sigma^2_m, i.e., for every integer m one can decide whether a given regular language is definable by a two-variable first-order formula with negation nesting depth at most m. More precisely, for every m we give omega-terms U_m and V_m such that an FO^2-definable language is in Sigma^2_m if and only if its ordered syntactic monoid satisfies the identity U_m \leq V_m.