First-Order Logic and Its Infinitary Quantifier Extensions over Countable Words

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