Two conjectures of Conway on rational expressions of languages are proved: the two following systems of identities are complete systems of identities (i.e. each rational identity is a consequence of the system):
1. The identities , and for each finite monoid .
2. The identities , and for each finite group .
There special identities are: ; ; , where is an alphabet in bijection with , the natural monoid homomorphism, and represents a rational expression naturally associated to this language.
The considerable work done by the author in order to solve these conjectures has many byproducts: completeness of certain meta-rule systems; characterization fo aperiodic semigroups by deductibility of their rational expression from and ; formal proof of Schützenberger’s star-free theorem; deduction of the matrix semigroup identity from the semigroup identity; stability of identities under operations (subsemigroup, quotient, semidirect product), which allows to use the theorem of Krohn-Rhodes; completeness of , together with the symmetric group identities.