  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.
