The alternation hierarchy in two-variable first-order logic FO 2 [<] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. We consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment Σ 2 m of FO 2 is defined by disallowing universal quantifiers and having at most m − 1 nested negations. The Boolean closure of Σ 2 m yields the m th level of the FO 2 -alternation

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... hy. We give an effective characterization of Σ 2 m , i.e., for every integer m one can decide whether a given regular language is definable in Σ 2 m . Among other techniques, the proof relies on an extension of block products to ordered monoids.