λ-Definability on free algebras

Abstract

Zaionc, M., λ-Definability on free algebras, Annals of Pure and Applied Logic 51 (1991) 279-300. A λ-language over a simple type structure is considered. There is a natural isomorphism which identifies free algebras with nonempty second-order types. If A is a free algebra determined by the signature S A = [α 1,...,α n ], then by a type τ A we mean τ 1,...,τ n →0 where τ i =0 α i →0. It can be seen that closed terms of the type τ A reflex constructions in the algebra A. Therefore any term of the type (τ A ) n →τ A defines some n-ary mapping in algebra A. The problem is to characterize λ-definable mappings in any free algebra. It is proved that the set of λ-definable operations is the minimal set that contains constant functions and projections and is closed under composition and limited recursion. This result is a generalization of the result of Schwichtenberg (1975) and Statman (1979) which characterize the λ-definable functions over the natural number type (0→0)→(0→0), i.e algebra [1, 0], as well as of the result of Zaionc (1987) for λ-definable word operations over type (0→0) n →(0→0), i.e algebra [1,...,1,0], and of the results about λ-definable tree operations (Zaionc, 1988 and 1990), i.e in algebra [2, 0]. Some of the examples in Section 5 are based on a publication of Zaionc (1988).