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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 ≦∗B⇒A B, yet not conversely.
(2) A + B ∈ ΛR⇒A 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 ∈ ΛR⇒min(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].
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