I started tradeing cryptocurrencies and while I was thinking about some exchange rates I started wondering about the following:

Let $ C_{i}$ be $ n$ currencies that can be exchanged with the rates $ R_{ij}$ and the following relation holds:

$ V_{i}R_{ij}=V_{j}$ where $ V_{i}$ is a certain value in currency $ i$ , therefor $ R_{ii}=1$ .

For example $ R_{$ £}=0.7382£/$ $ and therefor $ 100 $ * 0.7382 £/$ = 73.82 £ $

In a exchange world with no fees there would be the following relation:

$ R_{ij}= 1/R_{ji}$

in reality there is no such system, there are always fees but the exchange rates are always very close so:

$ R_{ij}=1/R_{ji}-f$

where $ f$ is a fee in the range between 0.01 and 0.1 of $ Ci$ .

What I described there is what I was able to figure out myself about currency exchange rates and a formal description of it. I am not a mathematician but an engineer and am able to comprehend higher mathematics if explained reasonably. I wondered why there is no “perpetuum moneyle”, namely a loop where you exchange at least three or more currencies to end up with your starting currency but with more value? Such a system would be able to generate value from nothing. My question is what, mathematically, prevents those loops who can print money from forming. Or better formulated:

1. Under what conditions there is no loop $ L= R_{12}*R_{23}*…*R_{n1}$ where $ L>1$ ?

2. Are there cases where this is possible?