Like $N’s$ – a point of view on natural numbers

Abstract

We define and study some simple structures which we call likens and which are conceptually near to both sets of natural numbers, i.e. $N$ with addition and $N^* = N$ \ $\{0\}$ with multiplication. It appears that there are many different likens, which makes it possible to look on usual natural numbers from a more general point of view. In particular, we show that $N$ and $N^*$ are related to some functionals on the space of likens. A similar idea is known for a long time as the Beurling generalized numbers. Our approach may be considered as a little more natural and more general, since it admits the finitely generated likens.