Locally trivial categories and unambiguous concatenation

Abstract

We use the recently developed theory of finite categories and the two-sided kernel to study the effect of the unambiguous concatenation product of recognizable languages on the syntactic monoids of the languages involved. As a result of this study we obtain an algebraic characterization (originally due to Pin) of the closure of a variety of languages under boolean operations and unambiguous concatenation, and a new proof of a theorem of Straubing characterizing the closure of a variety of languages under boolean operations and concatenation. We also note some connections to the study of the dot-depth hierarchy.