Realizability of prescribed numbers of non-atomic molecules in numerical monoids

Let NN be a numerical monoid, and let M(N)\mathcal{M}(N) denote its set of molecules and A(N)\mathcal{A}(N) its set of atoms.

Realizability conjecture. For every nN2n \in \mathbb{N}_{\ge 2} there exists a numerical monoid NN such that

M(N)A(N)=n.|\mathcal{M}(N) \setminus \mathcal{A}(N)| = n.

This asks which finite cardinalities can occur for the molecules of a numerical monoid that are not atoms. The statement is presented as a realizability conjecture motivating the paper; no resolution is supplied in the given text.

Sources & referencesView supporting material

Primary source

Marly Gotti and Marcos M. Tirador, “On the set of molecules of numerical and Puiseux monoids”, arXiv:2008.09904 (2021).

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