Two-Variable Logic over Countable Linear Orderings

We study the class of languages of finitely-labelled countable linear orderings definable in twovariable first-order logic. We give a number of characterisations, in particular an algebraic one in terms of circle monoids, using equations. This generalises the corresponding characterisation, namely variety DA, over finite words to the countable case. A corollary is that the membership in this class is decidable: for instance given an MSO formula it is possible to check if there is an equivalent two-variable logic formula over countable linear orderings. In addition, we prove that the satisfiability problems for two-variable logic over arbitrary, countable, and scattered linear orderings are Nexptime-complete. 1998 ACM Subject Classification F.4.3 Formal Languages