The problem of 0-classes
This is the section which contains current results on the pairing of
0-classes. The goal here is, given 0-classes K1,
K2, to determine whether or not another 0-class K exists
such that K2 is the disjoint union of classes isomorphic
to K1 and K2.
Let K be a 0-class.
A trunk-member of K is an x in K such that
for any n in N, there exists y such that fn(y) = x. The
trunk of K, denoted T(K), is the set of all trunk-members of K.
An element x of T(K) is called a node of T(K) (or a node of K) if
the cardinality of the set {y in T(K): f(y) = x} is greater than 1. We
denote this cardinal by d(x) and the set of all nodes by N(K) (or N(T(K))).
Let T be a trunk. Define a relation BT on N(T) where x
BT y iff
fn(x) = y for some n in N. We say in that case that x is
behind y.
Let T1 and T2 be trunks. We say that T1
is homeomorphic to T2 iff there exists a bijection
s:N(T1) -> N(T2) such that for x1,
x2 in N(T1), x1
BT1 x2 iff
s(x1) BT2 s(x2).
We turn to an important theorem involving these concepts, which
delivers insight into the pairing of 0-classes.
Theorem: Let T be a trunk such that there are no x, y in N(T) such that
f(x) = y. If T2 = T1 U T2 where
T1 and T2 are disjoint trunks, then T1
and T2 are homeomorphic.
Proof: It is easy to check that N(T1) U N(T2) is
the union of all pairs {x, f(x)} for x in N(T). (By hypothesis, these
sets are disjoint.) Moreover, if {a, b} is such a pair, and a is in
T1, then b is in T2. Define
s:N(T1) -> N(T2) where, if x from N(T1),
and {x, b} is one of the sets above, then s(x) = b. Then s is obviously
a bijection. Let {x1, s(x1)} and
{x2, s(x2)} be two of the pairs above. If
x1, x2 are in T1 and
x1 BT1 x2, so
fn(x1) = x2, then we must have one of the
following cases:
f(x1) = s(x1) and f(x2) = s(x2):
then fn(s(x1) = s(x2)
f(s(x1)) = x1 and f(x2) = s(x2):
then fn + 1(s(x1)) = s(x2)
f(x1) = s(x1) and f(s(x2)) = x2):
then fn - 1(s(x1)) = s(x2)
f(s(x1)) = x1 and f(s(x2)) = x2):
then fn(s(x1)) = s(x2)
In either of these cases, s(x1) BT2
s(x2), meaning s is the desired bijection and T1 is
homeomorphic to T2. QED
Previous
Next
Back to Table of Contents
Mail me
Back to my Homepage
