By Alt R., Vignes J.
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Alk ), σ) = (1) (ϕl ((a11 , . . , a1k ), . . , (al1 , . . , alk ), σ), . . , (k) ϕl ((a11 , . . , a1k ), . . , (al1 , . . , alk ), σ)). (i) If every ϕl (i = 1, . . , k) may depend on a11 , . . , a1i−1 , . . , al1 , . . , ali−1 and σ only, then we speak of a cascade generalized product. A similar generalization of the cascade product has been introduced in . Now let us extend ϕl to arbitrary nonnegative integer m and mapping ϕm from (A1 × . . × Ak )m × TΣ (Ξm ) to TΣ (1) (Ξm ) × .
3) A is a ranked alphabet consisting of unary operators, the state set of A. It is assumed that A is disjoint with all other sets in the deﬁnition of A, except A. 4) A ⊆ A is the set of ﬁnal states. 5) P is a ﬁnite set of productions of the following two types: i) x → a(q) (x ∈ Xm , a ∈ A, q ∈ TΩ (Yn )), ii) σ(a1 , . . , al ) → a(q(ξ1 , . . , ξl )) (σ ∈ Σl , l > 0, a1 , . . , al , a ∈ A, q(ξ1 , . . , ξl ) ∈ TΩ (Yn ∪ Ξl )). If a ∈ A is a state and p is a tree, then we generally write ap for a(p).
Recall that the polynomial closure of a class of languages C is the class of languages which are union of products of languages of C. Proposition 5 shows that decomposable languages are closed under polynomial closure. In , Schnoebelen proposed the following conjecture. Conjecture 1. A language is decomposable if and only if it belongs to the polynomial closure of the commutative languages. Denote by Pol(Com) the polynomial closure of the commutative languages. This class contains in particular the ﬁnite and the coﬁnite languages.