  We study the family of rational sets of words, called completely reducible
and which are such that the syntactic representation of their characteristic
series is completely reducible. This family contains, by a result of
Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel
and Reutenauer, the cyclic sets. We study the closure properties of this
family. We prove a result on linear representations of monoids which gives a
generalization of the result concerning the complete reducibility of the
submonoid generated by a bifix code to sets called birecurrent. We also give a
new proof of the result concerning cyclic sets.
