12
GENERALIZATIONS OF THE PERRON-FROBENIUS THEOREM
Since
we conclude that
U ^
lAm+l,JA=j
ra+1 /i
S
j
A=l j = l 3
which shows Eq. (3.3).
REMARK
3.2. Assume that L = {i G Z | 1 i m + 1}, with the usual
ordering, that E is a set with n elements, that {6i : Z E | i G L} is an admissible
array on n symbols and that 6i has minimal period pi. Assume also that pi ^ pj
for 1 i j m -f 1 and define L by
L = {p* | 1 i m + 1}.
Then L inherits a total ordering "- from L by
p - q if and only if p = p^ and q pj and I i j m + l.
Thus we can use L as the totally ordered index set for the admissible array in
Theorem 3.1, rather than L. Also, if p = pi, we can define 6P = 6i and Bp =
{@i(j) I J' £ ^} a n ( i work with the admissible array {#p | p G L}.
Conversely, suppose we are given a set L of m + 1 distinct integers p n and
suppose that E is a set with n elements. Then L is array-admissible for n if and
only if we can find a total ordering - on L and maps 6P : Z + E, p G L, such that
l-Spl P? where B
p
= {0p(j) \ j G Z}, 0p has minimal period p, and {0p \ p G L}
is an admissible array on n symbols for the total ordering on L. Since our typical
object of study will be sets L which are potentially array-admissible for n, the above
viewpoint will be useful.
The following corollaries of Theorem 3.1 are useful in the sequel.
COROLLARY
3.3. Let L = {i
G
Z | 1 i m + 1} and E be a set with
|E| = n. For each i G L, let Oi be a periodic map of minimal period Pi n and
suppose that \Bi\ = Pi, where Bi {Oi(j) \ j G Z}. Assume that BPi f) BPj ^ 0 for
lijm + l and assume that Sj C L, 1 j fx, are pairwise disjoint sets
with L = Uj=i 8j- F°r each j , 1 j n, assume that there is an integer rj 1
such that gcd(p, q)\rj whenever p = Pi for i G Sj and q = Pk for k ^ i, k G L. Let
Z^
-^ 1, (3.16)
3 = 1
J
then there does not exist a permutation a of L such that {6a{i) \ i £ L} is an
admissible array on n symbols.
PROOF.
Suppose such a permutation a exists. If we write 0$ = 8a^ and
Bi = -Ba(i), our assumptions imply that {$i | i G L} is an admissible array on n
symbols and Bi H Bi+\ ^ 0 for 1 i m. If we define pi = pa^), Sj =
cr_1(5j),
Tj = rj and Sj = sj = |Sj|, our assumptions imply that whenever i G Sj,
gcd(pi-i,pi)\fj and g c d ^ p i + i ) ^ .
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