  We contribute to the refined understanding of the language-logic-algebra
interplay in the context of first-order properties of countable words. We
establish decidable algebraic characterizations of one variable fragment of FO
as well as boolean closure of existential fragment of FO via a strengthening of
Simon's theorem about piecewise testable languages. We propose a new extension
of FO which admits infinitary quantifiers to reason about the inherent
infinitary properties of countable words. We provide a very natural and
hierarchical block-product based characterization of the new extension. We also
explicate its role in view of other natural and classical logical systems such
as WMSO and FO[cut] - an extension of FO where quantification over
Dedekind-cuts is allowed. We also rule out the possibility of a finite basis
for a block-product based characterization of these logical systems. Finally,
we report simple but novel algebraic characterizations of one variable
fragments of the hierarchies of the new proposed extension of FO.

Logic in Computer Science;Formal Languages and Automata Theory