Vol. 16, No. 3, 1966

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Two notes on regressive isols

Joseph Barback

Vol. 16 (1966), No. 3, 407–420

This paper deals with regressive functions and regressive isols. It was proven by J. C. E. Dekker in [2] that the collection ΛR of all regressive isols is not closed under addition. In the first note of this paper we shall given another proof of this fact by considering a new relation, denoted by ∗∨, between infinite regressive isols. Let A and B denote infinite regressive isols. The main results established in the first note are:

(1) A BA∗∨B, yet not conversely.

(2) A + B ΛRA∗∨B, yet not conversely.

(3) There exist infinite regressive isols which are not ∨∗ related.

(4) ΛR is not closed under addition.

In addition, the following result is stated.

(5) A + B ΛRmin(A,B) A + B, yet not conversely.

In the second note we consider the relation between regressive isols. A natural question concerning this relation is whether A B, where A and B are regressive isols, is a necessary or a sufficient condition for the sum A + B to be regressive. In the second note we show that this condition is neither necessary nor sufficient.

We shall assume that the reader is familiar with the notations, terminology and main results of [1] and [2].

Mathematical Subject Classification
Primary: 02.70
Received: 5 October 1964
Published: 1 March 1966
Joseph Barback