Green’s relations form a fundamental tool in the analysis of the structure of semigroups and monoids, in particular for the finite ones. Understanding Green’s relations can give remarkable intuitions concerning these objects, and as a consequence for understanding the structure of regular languages of words (finite or infinite). In particular, this description generalizes the understanding of automata through their decomposition into strongly connected components that is used in many proofs.

During this talk, I will present some of the key results concerning Green’s relations, emphasizing on how one can think about regular languages using them. I will also show how these can be used for deriving non-trivial automata related results (such as McNaughton’s determinization result, Schützenberger’s characterization of star-free languages, Simon’s factorization forest theorem, ... though unfortunately I will not have time do to all these).