Möbius inversion on posets

Work first with a finite poset and functions taking values in the rationals or reals. The zeta kernel has value one on every comparable interval. Its convolution with another interval kernel sums over intermediate elements between the endpoints. A linear ordering compatible with the partial order represents this operation by upper-triangular matrices with diagonal ones, so it is invertible. This is the finite algebra behind Möbius inversion, with no analytic limiting process involved.

Define the Möbius kernel recursively: its diagonal values are one, and on a strict interval its value is minus the sum of the values from the lower endpoint to all earlier elements of that interval. The recursion ensures that summing these coefficients over any nontrivial interval gives zero. On a locally finite infinite poset the same interval recursion remains meaningful, because each interval is finite. However a cumulative sum over all predecessors also needs to be finite or otherwise justified; local finiteness alone does not guarantee finite lower sets.

Suppose $F(x)=\sum_{y\le x}f(y)$ on a finite poset. Substitute this expression into a proposed inverse sum with coefficients $\mu(y,x)$ and interchange the finite sums. The coefficient multiplying a fixed local value $f(z)$ becomes the sum of $\mu(y,x)$ over intermediate y from z to x. Because the Möbius kernel is the two-sided inverse of zeta, that coefficient is one when z=x and zero otherwise. Thus all unwanted cumulative contributions cancel.

On the subset poset, an interval from T to S is itself a Boolean lattice on the elements in S but not T. Its Möbius value is minus one to the size of that difference. The binomial theorem verifies the required cancellation: the alternating sum over all subsets of a nonempty difference is zero. The familiar signs in inclusion-exclusion therefore arise from the interval structure, not from a universal rule that every poset uses alternating rank signs. Other posets have different Möbius coefficients, including zero values.

For positive integers ordered by divisibility, the interval from one to n contains the divisors of n. The resulting arithmetic Möbius function is zero if n contains a squared prime factor, and otherwise is minus one to the number of its distinct prime factors. This follows from the recursion and cancellation over subsets of the distinct prime divisors. It is not the same as alternating by the number of divisibility steps in an arbitrary drawn chain.

To use inversion responsibly, identify what a local contribution means and what cumulative quantity sums those contributions. An index order on a list is not necessarily the relevant partial order; subset containment and divisibility organize different overlaps. Once the order is fixed, calculate small interval values to check orientation: a lower-sum formula needs coefficients from the lower argument to the upper one. Reversing the order changes the formula. A correct-looking alternating sum with the wrong comparable pairs can recover entirely different data.