<abstract xmlns="http://www.w3.org/1999/xhtml"> 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 ∑an of real terms, the p-rearranged series ∑ap(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. </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 ∑an of real terms, the p-rearranged series ∑ap(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.

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

Institute of Mathematics Silesian University of Technology Kaszubska 23 PL–44-100 Gliwice POLAND

Institute of Mathematics Silesian University of Technology Kaszubska 23 PL-44-100 Gliwice POLAND

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In the paper we present the selected properties of composition relation of the convergent and divergent permutations connected with commutation. We note that a...

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

Published Online: Aug 15, 2014

Page range: 13 - 22

Received: Feb 07, 2013

DOI: https://doi.org/10.2478/tmmp-2014-0002

Keywords
commutation of composition of permutations, convergent permutations, divergent permutations

© 2014

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

Published Online: Aug 15, 2014

Page range: 13 - 22

Received: Feb 07, 2013

DOI: https://doi.org/10.2478/tmmp-2014-0002

Keywordscommutation of composition of permutations, convergent permutations, divergent permutations

© 2014

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
commutation of composition of permutations, convergent permutations, divergent permutations