The (classical) Möbius function is the unique function that obeys the classical Möbius inversion formula:

Proposition 1 (Classical Möbius inversion)Let be functions from the natural numbers to an additive group . Then the following two claims are equivalent:

- (i) for all .
- (ii) for all .

There is a generalisation of this formula to (finite) posets, due to Hall, in which one sums over chains in the poset:

Proposition 2 (Poset Möbius inversion)Let be a finite poset, and let be functions from that poset to an additive group . Then the following two claims are equivalent:(Note from the finite nature of that the inner sum in (ii) is vacuous for all but finitely many .)

- (i) for all , where is understood to range in .
- (ii) for all , where in the inner sum are understood to range in with the indicated ordering.

Comparing Proposition 2 with Proposition 1, it is natural to refer to the function as the Möbius function of the poset; the condition (ii) can then be written as

*Proof:*If (i) holds, then we have for any . Iterating this we obtain (ii). Conversely, from (ii) and separating out the term, and grouping all the other terms based on the value of , we obtain (1), and hence (i).

In fact it is not completely necessary that the poset be finite; an inspection of the proof shows that it suffices that every element of the poset has only finitely many predecessors .

It is not difficult to see that Proposition 2 includes Proposition 1 as a special case, after verifying the combinatorial fact that the quantity

is equal to when divides , and vanishes otherwise.I recently discovered that Proposition 2 can also lead to a useful variant of the inclusion-exclusion principle. The classical version of this principle can be phrased in terms of indicator functions: if are subsets of some set , then

In particular, if there is a finite measure on for which are all measurable, we haveOne drawback of this formula is that there are exponentially many terms on the right-hand side: of them, in fact. However, in many cases of interest there are “collisions” between the intersections (for instance, perhaps many of the pairwise intersections agree), in which case there is an opportunity to collect terms and hopefully achieve some cancellation. It turns out that it is possible to use Proposition 2 to do this, in which one only needs to sum over chains in the resulting poset of intersections:

Proposition 3 (Hall-type inclusion-exclusion principle)Let be subsets of some set , and let be the finite poset formed by intersections of some of the (with the convention that is the empty intersection), ordered by set inclusion. Then for any , one has where are understood to range in . In particular (setting to be the empty intersection) if the are all proper subsets of then we have In particular, if there is a finite measure on for which are all measurable, we have

Using the Möbius function on the poset , one can write these formulae as

and
*Proof:* It suffices to establish (2) (to derive (3) from (2) observe that all the are contained in one of the , so the effect of may be absorbed into ). Applying Proposition 2, this is equivalent to the assertion that

Example 4If with , and are all distinct, then we have for any finite measure on that makes measurable that due to the four chains , , , of length one, and the three chains , , of length two. Note that this expansion just has six terms in it, as opposed to the given by the usual inclusion-exclusion formula, though of course one can reduce the number of terms by combining the factors. This may not seem particularly impressive, especially if one views the term as really being three terms instead of one, but if we add a fourth set with for all , the formula now becomes and we begin to see more cancellation as we now have just seven terms (or ten if we count as four terms) instead of terms.

Example 5 (Variant of Legendre sieve)If are natural numbers, and is some sequence of complex numbers with only finitely many terms non-zero, then by applying the above proposition to the sets and with equal to counting measure weighted by the we obtain a variant of the Legendre sieve where range over the set formed by taking least common multiples of the (with the understanding that the empty least common multiple is ), and denotes the assertion that divides but is strictly less than . I am curious to know of this version of the Legendre sieve already appears in the literature (and similarly for the other applications of Proposition 2 given here).

If the poset has bounded depth then the number of terms in Proposition 3 can end up being just polynomially large in rather than exponentially large. Indeed, if all chains in have length at most then the number of terms here is at most . (The examples (4), (5) are ones in which the depth is equal to two.) I hope to report in a later post on how this version of inclusion-exclusion with polynomially many terms can be useful in an application.

Actually in our application we need an abstraction of the above formula, in which the indicator functions are replaced by more abstract idempotents:

Proposition 6 (Hall-type inclusion-exclusion principle for idempotents)Let be pairwise commuting elements of some ring with identity, which are all idempotent (thus for ). Let be the finite poset formed by products of the (with the convention that is the empty product), ordered by declaring when (note that all the elements of are idempotent so this is a partial ordering). Then for any , one has where are understood to range in . In particular (setting ) if all the are not equal to then we have

Morally speaking this proposition is equivalent to the previous one after applying a “spectral theorem” to simultaneously diagonalise all of the , but it is quicker to just adapt the previous proof to establish this proposition directly. Using the Möbius function for , we can rewrite these formulae as

and
*Proof:* Again it suffices to verify (6). Using Proposition 2 as before, it suffices to show that

## 6 comments

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23 February, 2021 at 6:57 pm

Sam HopkinsYour propositions feel very similar to “Rota’s crosscut theorem.” See for instance Section 3.9 of Stanley’s EC1, available at http://www-math.mit.edu/~rstan/ec/ec1/.

26 February, 2021 at 10:43 pm

Aditya Guha RoyI had a similar feeling at first glance. But proposition 6 is actually not exactly the same as the crosscut theorem; it deals with a different treatment.

24 February, 2021 at 6:17 am

allenknutsonMy favorite statement about Möbius inversion is the computation of the Möbius function for a simplicial complex (i.e. the coefficients needed when trying to write the function 1 as a linear combination of the characteristic functions of the closed simplices). It says: the coefficient of a face is 1 – the Euler characteristic of the link of that face.

When the simplicial complex is a (shellable) ball, the links of interior faces are spheres, giving the coefficient (-1)^codimension. Whereas the links of exterior faces are hemispheres, giving the coefficient 0.

24 February, 2021 at 12:53 pm

AnonymousIs it true that since only finitary operations (e.g. finite sums) are involved, all proofs of such results can be based on only formal(!) arguments?

26 February, 2021 at 10:58 pm

Aditya Guha RoyIn trying to extend the idea to infinite posets you may need additional conditions to deal with the infinite products and sums which you’ll encounter; for instance if you look at Proposition 2, then you can see how the sum in ii can bother you if you try to extend the idea to deal with an infinite poset.

25 February, 2021 at 8:42 am

Oliver KnillHere is a topological angle to the Moebius picture (complementing Allen Knutsen’s remark): a poset P has as its Barycentric refinement the finite abstract simplicial complex G in which the elements are the non-empty subsets of P. For a subset X of G, the number chi(X) = sum_{x in X} omega(x) is known as the Euler characteristic of X, where omega(x) = (-1)^dim(x), dim(x)=|x|-1 and |x| is cardinality. With the core W^-(x)={y, y x } one can write the Moebius function as mu(x,y) =chi(W^+(x) \cap W^-(y)). (I like to think of W^+ and W^- as unstable and stable manifolds and W^+ cap W^- as a heteroclinic point The Hall equations then read f(x) = sum_{y<x} g(y), g(x) = sum_{y A to a ring A, define chi(X)=sum_{x in X} h(x) and the symmetric matrices L(x,y)=chi(W^-(x) cap W^-(y)) g(x,y) = omega(x) omega(y) chi(W^+(x) cap W^+(y)). The later can be seen as Green functions because g is the inverse of the Laplacian L in the case if h takes values in the units One should think of g(x,y) as the potential energy between x and y. As V(x) = sum_y g(x,y) is a potential or index, Poincare-Hopf then is chi(G) = sum_{x,y in G} g(x,y). This so far only uses the additive structure of A. If A is a ring, then det(L)=det(g)=prod_{x in G} h(x). In the topological case h(x)=omega(x), the number theoretically interesting h(x)=1 or if h(x) takes values in the unit circle of the complex plane, L and g^* are unimodular matrices which are inverses of each other. For h(x)=1, one gets positive definite integral quadratic forms which are inverses of each other. Inclusion-exclusion is used in the proof https://arxiv.org/abs/2010.09152 .