The strictness of IRS can be obtained by just removing the label
from
,
and therefore by increasing by one at most
its compactness (or linearity compactness). Because a cyclic interval
is the union of two linear intervals, we have the trivial collection
of inclusions:
In the following,
means that
and
.
Because in [FJ88] and in [BvLT91] it is shown
-
but not in k-
,
we have:
Theorem 57, and 58 can be improved by Theorem 47.
From Theorem 47, for every ,
-
-
,
and also
-
.
From the second point of Theorem 47,
-
implies that
.
Hence
-
.
It follows that k-
-
and k-
-
.
Similarly
-
,
but
-
,
and
-
.