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We obtain a logical characterization of an important class of regular languages, denoted ${\mathcal DO}$, and of its most important subclasses in terms of two-variable sentences with ordinary and modular quantifiers but in which all modular quantifiers lie outside the scope of ordinary quantifiers. The result stems from a new decomposition of the variety of monoids DO in terms of iterated block products. This decomposition and the ensuing logical characterization allows us to shed new light on recent results on regular languages which are recognized by bounded-depth circuits with a linear number of wires and regular languages with small communication complexity.

Restricted Two-Variable FO + MOD Sentences, Circuits and Communication Complexity