It is shown that the syntactic algebra of the characteristic series of a rational language L is semisimple in the following two cases: L is a free submonoid generated by a bifix code, or L is a cyclic language.

This chapter has two appendices, one on semisimple algebras (without proofs) and another on simple semigroups, with concise proofs. We use the symbols A1 and A2 to refer to them.

Bifix codes

Let E be a set of endomorphisms of a finite dimensional vector space V. Recall that E is called irreducible if there is no subspace of V other than 0 and V itself which is invariant under all endomorphisms in E. Similarly, we say that E is completely reducible if V is a direct sum V = V1 ⊕…⊕ Vk of subspaces such that for each i, the set of induced endomorphisms e∣Vi of Vi, for e ϵ E, is irreducible.

A set of matrices in Kn×n (K being a field) is irreducible (resp. completely reducible) if it is so, viewed as a set of endomorphisms acting at the right on K1×n, or equivalently at the left on Kn×1 (for this equivalence, see Exercises 1.1 and 1.2).

A linear representation (λ, μ, γ)ofaseries S ϵ K⟪A⟫ is irreducible (resp. completely reducible) if the set of matrices {μa ∣ a ϵ A}(or equivalently the sets μA* or μ(K⟨A⟩)) is so.