We study the Möbius function and the Möbius inversion theorem on posets. The latter is an important tool in inverting certain types of sums and with that a useful way of enumerating. The Möbius function has interesting characteristics that make computing it much easier. It is particularly fascinating on lattices. We study both the theory and applications on a few cases of lattices. We find that the Möbius inversion on natural numbers with the divisibility ordering gives us none other than the classical Möbius inversion from number theory. On the other hand, we can use its application on the power set to prove the very important inclusion--exclusion principle. The Möbius function can be used to define the Euler characteristic of a poset, which we then connect to the usual topological concept.
In the work we also study Whitney numbers, their connection to the Möbius function and how we can view them in light of Stirling numbers, if we apply them to the set of all partitions of a finite set. Using the Möbius inversion, we also give a representation of Stirling numbers with the Möbius function.
|
