Relational morphisms, transductions and operations on languages

The aim of the article is to present two algebraic tools (the representable transductions and the relational morphisms) that have been used in the past decade to study operations on recognizable languages. This study reserves a few surprises. Indeed, both concepts were originally introduced for other purposes : representable transductions are a formalization of automata with output and have been mainly studied in connection with the theory of context-free languages, while relational morphisms were introduced by Tilson to solve some problems related to the wreath product decomposition of finite semigroups. But it turns out that relational morphisms are a very powerful tool in the study of recognizable languages and that transductions lead to some very nice problems on finite semigroups. Eilenberg’s variety theorem gives a one-to-one correspondence between varieties of semigroups and varieties of languages. Part of the results reviewed in this article show that, in certain cases, this correspondence can be extended to operations. That is, an operation on languages (such as concatenation, lengthpreserving morphism, etc.) is in correspondence with an operation on semigroups. It is therefore tempting to ask whether the most natural operations on languages (respectively semigroups) have a natural counterpart in terms of semigroups (respectively languages). This leads to a number of difficult problems, some of which are still unsolved.