Duality and the equational theory of regular languages

On one hand, the Eilenberg variety theorem establishes a bijective correspondence between varieties of formal languages and varieties of finite monoids. On the other hand, the Reiterman theorem states that varieties of finite monoids are exactly the classes of finite monoids definable by profinite equations. Together these two theorems give a structural insight in the algebraic theory of finite automata. We explain how duality theory can account for the combination of this two theorems, as it was pointed out by (Gehrke, Grigorieff, and Pin, 2008). The theory of formal languages is basically concerned with the description of properties of sequences, which are nothing else than sets of sequences. For any finite non empty set A of symbols called an alphabet, we define a (formal) language as a set of finite sequences of symbols in A. In this context, finite sequences are called words and are naturally endowed with the concatenation operation. The set of all words on the alphabet A forms the free monoid A