Algebraic characterizations and block product decompositions for first order logic and its infinitary quantifier extensions over countable words

We contribute to the refined understanding of language-logic-algebra interplay in a recent algebraic framework over countable words. Algebraic characterizations of the one variable fragment of FO as well as the boolean closure of the existential fragment of FO are established. We develop a seamless integration of the block product operation in the countable setting, and generalize well-known decompositional characterizations of FO and its two variable fragment. We propose an extension of FO admitting infinitary quantifiers to reason about inherent infinitary properties of countable words, and obtain a natural hierarchical block-product based characterization of this extension. Properties expressible in this extension can be simultaneously expressed in the classical logical systems such as WMSO and FO[cut]. We also rule out the possibility of a finite-basis for a block-product based characterization of these logical systems. Finally, we report algebraic characterizations of one variable fragments of the hierarchies of the new extension.