Block products for algebras over countable words and applications to logic

We propose a seamless integration of the block product operation to the recently developed algebraic framework for regular languages of countable words. A simple but subtle accompanying block product principle has been established. Building on this, we generalize the well-known algebraic characterizations of first-order logic (resp. first-order logic with two variables) in terms of strongly (resp. weakly) iterated block products. We use this to arrive at a complete analogue of Schutzenberger-McNaughton-Papert theorem for countable words. We also explicate the role of block products for linear temporal logic by formulating a novel algebraic characterization of a natural fragment.