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

Abstract

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.