On countable chains having decidable monadic theory (Q2892678)

This is a contribution to a still open problem raised by \textit{C. C. Elgot} and \textit{M. O. Rabin} in [J. Symb. Log. 31, 169--181 (1966; Zbl 0144.24501)] -- asking whether there exist maximal decidable structures, i.e., structures \({\mathfrak M}\) with a decidable first-order (FO) theory and such that the FO-theory of any expansion of \({\mathfrak M}\) by a non-definable predicate is undecidable -- in the framework of monadic second-order logic. The authors show that if the monadic second-order theory of a countable chain \(C\) is decidable, then \(C\) has a non-trivial expansion with decidable monadic second-order theory (a chain is here an expansion of a linear order by monadic predicates).