Abstract
We have seen that, for the most part, natural operations on (varieties of) languages corrspond to natural operations on (varieties of) monoids: concatenation corresponds to aperiodic relational morphisms and to Schutzenberger products, non ambiguous concatenation to locally trivial relational morphisms, length preserving morphisms to power monoids, sequential functions to wreath products, etc. In some cases, the corresponding operation is still unknown: for instance, it would be interesting to find an operation on languages corresponding to nilpotent relational morphisms,a nd an operation on monoids corresponding to shuffle. This correspondence between languages and monoids has motivated numerous research articles, and a number of problems are still open: on the semigroup side, the complete classification of the varieties of the form PV or the (many) decidability problems about varieties of the form V-1W or V*W; on the language side, the characterisation of varieties closed under shuffle or under pure star, and all problems related to the star operation and to the concatenation product (see the articles of P. Weil and K. Hashiguchi in this volume).
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