Sciences is gratefully acknowledged.

Abstract. There exists an infinite ascending chain of finitely generated clones on a nine-element set for which the corresponding algebras are alternately finitely based and inherently nonfinitely based. In 1981, at the Chautauqua resort near Boulder, Colorado, in the presence of a large walnut salad, G. McNulty posed the following problem (published in [5, p. 227]): Problem. Let A be a finite set with at least three elements. In the lattice of clones over A, is there always an infinite ascending chain C0, C1,... of finitely generated clones such that 〈A,Ci 〉 is finitely based when i is even and not finitely based when i is odd? McNulty was motivated to pose this problem by the fact that a negative answer might lead to a positive solution to a long-standing finite basis problem of Tarski. In the meantime, R. McKenzie has given a negative solution to Tarski’s problem [3] and so one might suspect that the Chautauqua Problem has a positive answer. Until recently, however, it was not known whether there exists any finite set A with the Chautauqua property. In this paper we prove Theorem 1. On a nine-element set A it is possible to define an infinite sequence of operations Fn (n < ω) so that if An = 〈A, (Fi: i ≤ n)〉, then for all k < ω, A2k is finitely based and every subdirectly irreducible member of HSP(A2k) is in HS(A2k), while A2k+1 is inherently nonfinitely based and HSP(A2k+1) is residually large. This provides a strong affirmative answer to McNulty’s question when restricted to nine-element sets. The theorem can be easily extended to any set with at least nine elements. McNulty subsequently observed that with more work, Theorem 1 can be extended to any set having at least three elements, fully solving the Chautauqua problem. (The latter result may eventually appear in [6].) Our proof of Theorem 1 combines several components of McKenzie’s refutation of the RS conjecture and our finite basis theorem for residually finite varieties with a semilattice term operation