Ed-regressive sets of order n

Abstract

A number is a nonnegative integer, and E is the set of numbers. In [3], J. C. E. Dekker introduced the concept of a regressive set of order n as the range of a one-one function f of n arguments such that (i) domain f ⊆ En and, if (x1, …, xn) ∈ domain f, and yi ≤ ≤ xi for 1 ≤ i ≤ n, then (y1, …, yn) ∈ domain f, and (ii) if 1 ≤ i ≤ n and (x1, …, xn) ∈ domain f, then f(x1 … xi−1, xi ∸ 1, xi+1 … xn) can be found effectively from f(x1 … xn). (0 ∸ 1 = 0 and, for m ≥ 1, m ∸ 1 = m − 1.) Since one can take the view, as Dekker did when first introducing regressive functions in [1], that a regressive set of order one is the range of a function of the above type which is of order one and everywhere defined, it seems natural to study the n-dimensional analogue in which (i) is replaced by “domain f = En.” It is the purpose of this paper to study such sets.