The product of rational languages

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Abstract

The very basic operation of the product of rational languages is the source of some of the most fertilizing problems in the Theory of Finite Automata. Indeed, attempts to solve McNaughton's star-free problem, Eggan's star-height problem and Brzozowski's dot-depth problem, all three related to the product, already led to many deep and ever expanding connections between the Theory of Finite Automata and other parts of Mathematics, such as Combinatorics, Algebra, Topology, Logic and even Universal Algebra. We review some of the most significant results of the area, obtained during the last 35 years, and try to show their contribution to our understanding of the product.