Mobius Function for Posets

we recall the classical facts about the Mobius function for posets. As mentioned in Section 10.4, for an arbitrary poset P, let I P) denote the set of the finite intervals of P. Define the function p I P) — C as follows  

The function /x is called the Mobius function. An alternative way to define y, is via the incidence algebra of P. All functions / I P) — C form a C-algebra, with pointwise addition, pointwise multiplication by complex numbers, and the following convolution product for /, I P) — C we define 

Theorem 10.24. For any finite poset P with a maximal and minimal elements, we have 

On the other hand, topologically A P) is obtained from A Q) by attaching a cone with apex in x and base 4(6, x). In terms of Euler characteristics we 

The comparison of (10.17) with (10.18), together with the induction assumption, yields the result.  

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The Mobius function for posets is in fact a special case of the notion of a Mobius function for acyclic categories. 

Originally, the Mobius function was introduced for integers. The textbook [Sta97] contains a great deal of material about the Mobius function for posets, including Theorem 10.24, which is also called Hall s Theorem. 

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