The Pseudofinite Monadic Second Order Theory of Linear Order

Monadic second order logic is the expansion of first order logic by quantifiers ranging over unary relations. We study the shared monadic second order theory of finite linear orders, i.e. the pseudofinite monadic second order theory of linear order, using a first order setup . We give explicit (and recursive) axioms and characterise the completions in terms of residue sequences. A connection with profinite algebra, in particular with the free profinite monoid on one generator, is established via extended Stone duality.