this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun.

We show that each level of the quantifier alternation hierarchy within FO²[<] on words is a variety of languages. We use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy – and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit, a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels.