  We develop an algebraic notion of recognizability for languages of words
indexed by countable linear orderings. We prove that this notion is effectively
equivalent to definability in monadic second-order (MSO) logic. We also provide
three logical applications. First, we establish the first known collapse result
for the quantifier alternation of MSO logic over countable linear orderings.
Second, we solve an open problem posed by Gurevich and Rabinovich, concerning
the MSO-definability of sets of rational numbers using the reals in the
background. Third, we establish the MSO-definability of the set of yields
induced by an MSO-definable set of trees, confirming a conjecture posed by
Bruy{\`e}re, Carton, and S{\'e}nizergues.
