On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)
Roman Wituła, Edyta� Hetmaniok, Damian Słota · Tatra Mountains Mathematical Publications · 2014
Abstract In the paper we present the selected properties of composition relation of the convergent and divergent permutations connected with commutation. We note that a permutation on ℕ is called the convergent permutation if for each convergent series ∑a n of real terms, the p-rearranged series ∑a p(n) is also convergent. All the other permutations on ℕ are called the divergent permutations. We have proven, among others, that, for many permutations p on ℕ, the family of divergent permutations q on ℕ commuting with p possesses cardinality of the continuum. For example, the permutations p on ℕ having finite order possess this property. On the other hand, an example of a convergent permutation which commutes only with some convergent permutations is also presented.