Fan–Tringali Conjecture on length sets in power monoids of N0

Determine whether, in both the finitary power monoid P_fin(N0) and the reduced finitary power monoid P_fin,0(N0) under set addition, every nonempty finite subset of the positive integers N_{>=2} occurs as a length set of some element.

Background

Power monoids are additive monoids whose elements are nonempty finite subsets of an ambient monoid (here M = N0) with the operation given by set addition. In this context, an atom is an indecomposable set, and the length set L(A) of an element A records all possible numbers of atoms in factorizations of A.

The paper recalls a conjecture by Fan and Tringali asserting that, for the finitary power monoid P_fin(N0) and its reduced variant P_fin,0(N0), every nonempty finite subset of N_{>=2} appears as a length set. This conjecture stands in contrast with a density result indicating that almost all elements are atoms (whose length set is {1}), highlighting the difficulty of realizing arbitrary finite subsets of N_{>=2} as length sets.

The present work advances toward this conjecture by determining the set of elasticities for these structures and showing full elasticity in relevant settings, thereby supporting the conjecture though not resolving it in full generality.