On commutation properties of the composition relation of convergent and divergent permutations PART I

Abstract

In the paper we present the selected properties of composition re-
lation of the convergent and divergent permutations connected with
commutation. We note that a permutation on $ \matbb{N} $i s called the conver-
gent permutation if for each convergent series

$\Sigma a_n$

of real terms the
$p$−rearranged series $\Sigma a_p(n)$ is also convergent.

All the other permu-
tations on $ \matbb{N} $ are called the divergent permutations. We have proven,
among others, that for many permutations p on $ \matbb{N} $ the family of diver-
gent permutations $q$ on $ \matbb{N} $, commuting with $p$, possesses cardinality of
the continuum. For example, the permutations $p$ on $ \matbb{N} $ having finite
order possess this property. For contrast, the example of convergent
permutation which commute only with some convergent permutations
is also presented.