We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels. 1 Introduction We study first-order sentences interpreted in finite words over a finite alphabet Σ, with the single relation < on positions in

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... he word. It is well known (Kamp [6], Immerman and Kozen [5]) that every such sentence is equivalent to one in which only three variables are used. There has been extensive study, from the standpoint of first-order and temporal logic, automata theory, and algebra, of the fragment F O 2 [<] of sentences that use only two variables. (See, for example, Ettesami, Vardi and Wilke [4]; Schwentick, Thérien and Vollmer [13]; Straubing and Thérien [16]. Tesson and Thérien [17] give a broad-ranging survey of the many places in which the class of languages definable in this logic arises.) Weis and Immerman [20] examined the hierarchy within F O 2 [<] based on alternation of quantifiers. Using model-theoretic methods, they showed that this hierarchy is strict. Kufleitner and Weil [9] show that each level of the hierarchy defines a variety of languages. This implies, among other things, that whether a regular language L ⊆ Σ * can be defined by a sentence of a given level k in the hierarchy is completely determined by the syntactic monoid M (L) of L. While they do not provide an explicit algebraic description of the levels, Kufleitner and Weil do show that one can effectively compute the alternation depth of a given language in F O 2 [<] with an error no more than 1. Here we give an exact algebraic characterization of each level of the alternation hierarchy; that is, we give an algebraic description of sequence V n of families of finite monoids with the property that L is defined by a sentence with k quantifier alternations if and only if M (L) ∈ V k. Our characterization is in terms of the two-sided semidirect product of finite monoids and of pseudovarieties of finite monoids. More precisely, we show that the k th level of the hierarchy corresponds to the weakly iterated two-sided semidirect product of k copies of the pseudovariety J of J-trivial monoids. While many algebraic decompositions of the pseudovariety DA corresponding to F O 2