One counter languages and the chevron operation

— For a language L and new symbols a and b, define the chevron of L as < L > = { a" wb | n ̂ 0, weL}. Thefamily ofone counter languages is strongly résistant to the chevron opération in the sensé that ( L } is a one counter language if and only if L is regular. Résumé. — Soit L un langage défini sur un alphabet ne contenant pas les lettres a et b. Alors, < L > = { a wb | n ̂ 0, weL) appartient à la famille des langages à un compteur si et seulement si L est un langage rationnel. The family of linear context-free languages not only is not closed under concaténation but is strongly résistant to concaténation in the following sensé. If Lx and L2 are languages over disjoint alphabets, then Lt L2 is linear contextfree only if either Li or L2 is regular [9], Goldstine showed that the least full semiAFL (family of languages containing at least one nonempty language and closed under union, homomorphism, inverse homomorphism, and intersection with regular sets) containing the 1-bounded languages has the same property [8], and recently Latteux demonstrated this property for the least full semiAFL containing the two-sided Dyck set on one letter [12], A similar phenomenon has been observed for other opérations. The family of ultralinear languages is strongly résistant to Kleene * in the sensé that, for a language L and a new symbol c, (Le)* is ultralinear if and only if L is regular [7]. The least full semiAFL containing the bounded languages is likewise strongly résistant to Kleene * [8]. We can make this concept more précise. For opérations on at least two languages, the définition of "strongly résistant" is obvious. DÉFINITION: Let O be a fe-ary opération on languages, k^ 2 and if a family of languages. We say that J5f is strongly résistant to 0> if, whenever (Llf . . . , Lk) is in J5f and the languages Lf are over pairwise disjoint alphabets, then there is somej such that L} is regular. (*) Reçu août 1978 et dans sa version définitive décembre 1978. (') The research reported in this paper was supported in part by the National Science Foundation under Grant No. MCS 78-04725. () Department of System Science, University of California, Los Angeles. R.A.I.R.O. Informatique théorique/Theoretical Informaties, 0399-0540/1979/189/$ 4.00 © Bordas-Dunod