<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper we continue our investigation concerning the concept of a <italic>liken</italic>. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in ℝ. The most important examples of likens are clearly the set of natural numbers ℕ with addition and the set of positive natural numbers ℕ* with multiplication, represented by the sequence <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2022-0007_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><mo>ln</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></mrow></math>
<tex-math>\left( {\ln \left( {n + 1} \right)} \right)_{n = 0}^\infty</tex-math>
</alternatives>
</inline-formula>. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this <italic>space of likens</italic> we consider elements up to isomorphism and define <italic>properties of likens</italic> as such that are isomorphism invariant. The main result of this paper is a theorem characterizing the liken ℕ* of natural numbers with multiplication in the space of all likens.</p>
</abstract>

In this paper we continue our investigation concerning the concept of a liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in ℝ. The most important examples of likens are clearly the set of natural numbers ℕ with addition and the set of positive natural numbers ℕ* with multiplication, represented by the sequence 


(ln(n+1))n=0∞
\left( {\ln \left( {n + 1} \right)} \right)_{n = 0}^\infty

. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this space of likens we consider elements up to isomorphism and define properties of likens as such that are isomorphism invariant. The main result of this paper is a theorem characterizing the liken ℕ* of natural numbers with multiplication in the space of all likens.

On a certain characterisation of the semigroup of positive natural numbers with multiplication

Department of Mathematics and Computer Science

Department of Mathematics and Natural Sciences

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

In this paper we continue our investigation concerning the concept of a liken. This notion has been defined as a sequence of non-negative real numbers, tending...

On a certain characterisation of the semigroup of positive natural numbers with multiplication

Published Online: Dec 08, 2022

Page range: 71 - 92

Received: May 09, 2022

Accepted: Oct 05, 2022

DOI: https://doi.org/10.2478/aupcsm-2022-0007

Keywords
Beurling numbers, distribution of prime numbers, Cauchy translation equation, numerical semigroups, Apéry sets

© 2022 Edward Tutaj, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

On a certain characterisation of the semigroup of positive natural numbers with multiplication

Published Online: Dec 08, 2022

Page range: 71 - 92

Received: May 09, 2022

Accepted: Oct 05, 2022

DOI: https://doi.org/10.2478/aupcsm-2022-0007

KeywordsBeurling numbers, distribution of prime numbers, Cauchy translation equation, numerical semigroups, Apéry sets

© 2022 Edward Tutaj, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
Beurling numbers, distribution of prime numbers, Cauchy translation equation, numerical semigroups, Apéry sets