bibkey: shala2024cosetlike
authors: Besfort Shala
year: 2024
title: "The Probabilistic Zeta Function of a Finite Lattice"
doi: null
url: https://arxiv.org/abs/2112.13766v6
claim: "Question 7.4 asks whether there exists a weakly coset-like atomistic lattice that is not strongly coset-like."
strata_touched:
  - D5/S3/Combinatorics/Lattices/ShalaCosetLikeAtomisticLattice
license: citation-only
triage: anchor

The probabilistic zeta function of a finite lattice

Verified locator

DOI: null

Source: https://arxiv.org/abs/2112.13766v6

Besfort Shala, “The Probabilistic Zeta Function of a Finite Lattice”, arXiv:2112.13766v6 (24 April 2023), published in Rocky Mountain Journal of Mathematics 54 (2024), 1511–1526.

Source statements

Equation (3.1) defines the probabilistic zeta function by

P(L, s) = ∑_{x ∈ L ∖ {0̂}} μ(x, 1̂) / [J : J_x]^s

where J is the set of join-irreducible elements and J_x = {j ∈ J : j ≤ x}.

Definition 5.1 says:

We say a lattice L is strongly coset-like if |J_x| divides |J| for every x ∈ L \ {0̂}.

Definition 5.2 says:

We say a lattice L is weakly coset-like if P(L, s) is a finite ordinary Dirichlet series.

Question 7.4, first sentence on page 16, asks:

Does there exist a weakly coset-like atomistic lattice that is not strongly coset-like?

The formal statement uses the universal negation of this existence question: every weakly coset-like atomistic finite lattice is strongly coset-like. The twelve-element lattice in the settlement refutes that universal statement and supplies an affirmative answer to the question. The d-divisible partition-lattice sentence, minimality of twelve elements, and exhaustive priority remain outside this note.