An effective characterization of the alternation hierarchy in two-variable logic

We give an algebraic characterization, based on the bilateral semidirect product of finite monoids, of the quantifier alternation hierarchy in two-variable first-order logic on finite words. As a consequence, we obtain a new proof that this hierarchy is strict. Moreover, by application of the theory of finite categories, we are able to make our characterization effective: that is, there is an algorithm for determining the exact quantifier alternation depth for a given language definable in two-variable logic.