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The Möbius function of a product poset is the product of the Möbius functions

Statement

Let $(P, \leq_{P})$ and $(Q, \leq_{Q})$ be locally finite posets. Define the product order on $P \times Q$ by

$(p, q) \leq (p^{'}, q^{'}) ⟺ p \leq_{P} p^{'} and q \leq_{Q} q^{'} .$

This relation is a partial order, the product poset is locally finite, and for comparable pairs

$μ_{P \times Q} ((p, q), (p^{'}, q^{'})) = μ_{P} (p, p^{'}) μ_{Q} (q, q^{'}) .$

Facts & Assumptions

Given: Locally finite posets $P, Q$ and elements $p \leq_{P} p^{'}$ , $q \leq_{Q} q^{'}$ .

[F1]

A partial order is reflexive, antisymmetric and transitive (Partial order and partially ordered set).

[F2]

Local finiteness means every closed interval is finite (Intervals in a poset; locally finite, lower-finite and upper-finite posets).

[L1]

A Cartesian product of finite sets is finite (The product rule: $∣ A \times B ∣ = ∣ A ∣ ∣ B ∣$ , and $∣ \prod_{i < m} A_{i} ∣ = \prod_{i < m} ∣ A_{i} ∣$ ).

[L2]

Finite Fubini interchanges a sum over a finite Cartesian product with its two iterated sums (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

[L3]

The Möbius function is the unique integer-valued function with diagonal value $1$ and vanishing interval sums off the diagonal (The Möbius recurrence: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ vanish when $x < y$ , The integers form a commutative ring).

Proof

technique · direct

1.1

The product relation is reflexive because both coordinate orders are reflexive; it is antisymmetric because two opposite product inequalities give equality in each coordinate; and it is transitive because coordinatewise inequalities compose. Hence it is a partial order.

1.2

Its intervals are exactly Cartesian products: $[(p, q), (p^{'}, q^{'})] = [p, p^{'}]_{P} \times [q, q^{'}]_{Q}$ . Both factors are finite by local finiteness, so the interval is finite by [L1]; thus $P \times Q$ is locally finite.

F2L1

1.3

Define $ν ((p, q), (p^{'}, q^{'})) : = μ_{P} (p, p^{'}) μ_{Q} (q, q^{'})$ . On the diagonal, $ν ((p, q), (p, q)) = 1 \cdot 1 = 1$ .

2.1

For a nontrivial product interval, finite Fubini gives $\sum_{(p, q) \leq (u, v) \leq (p^{'}, q^{'})} ν ((p, q), (u, v)) = (\sum_{p \leq_{P} u \leq_{P} p^{'}} μ_{P} (p, u)) (\sum_{q \leq_{Q} v \leq_{Q} q^{'}} μ_{Q} (q, v))$ .

step 1.2L2L3

3.1

Each factor in step 2.1 is $1$ when its endpoints agree and $0$ otherwise. Since the product interval is nontrivial, at least one coordinate pair has distinct endpoints, so the product is $0$ .

step 2.1L3

4.1

Thus $ν$ has the diagonal and recurrence properties of the Möbius function on $P \times Q$ , and uniqueness in [L3] gives $ν = μ_{P \times Q}$ .

step 1.3step 3.1L3∎

Depends on

Used by

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Sources

F. Gotti, Incidence Algebras, MIT 18.211 notes (standard reference, not scraped)
R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)