Theme and Variations on the Concatenation Product

The concatenation product is one of the most important operations on regular languages. Its study requires sophisticated tools from algebra, finite model theory and profinite topology. This paper surveys research advances on this topic over the last fifty years. The concatenation product plays a key role in two of the most important results of automata theory: Kleene’s theorem on regular languages [23] and Schützenberger’s theorem on star-free languages [60]. This article surveys the most important results and tools related to the concatenation product, including connections with algebra, profinite topology and finite model theory. The paper is organised as follows: Section 1 presents some useful algebraic tools for the study of the concatenation product. Section 2 introduces the main definitions on the product and its variants. The classical results are summarized in Section 3. Sections 4 and 5 are devoted to the study of two algebraic tools: Schützenberger products and relational morphisms. Closure properties form the topic of Section 6. Hierarchies and their connection with finite model theory are presented in Sections 7 and 8. Finally, new directions are suggested in Section 9.