Numerical semigroups from rational matrices III: semigroups of matricial dimension two and a counterexample to the lonely element conjecture

TL;DR

This paper characterizes certain numerical semigroups of matricial dimension two and provides a counterexample to a conjecture relating small elements being lonely to the semigroup's matricial dimension.

Contribution

It offers a complete characterization of semigroups of matricial dimension two and disproves a conjecture about the relationship between small elements and dimension.

Findings

Characterization of semigroups with matricial dimension two

Counterexample to the lonely element conjecture

Disproof of the conjecture relating small elements to dimension

Abstract

We characterize semigroups in $\{0,1,2,\ldots\}$ of matricial dimension $2$ and produce a counterexample to the conjecture that a numerical semigroup whose small elements are lonely has matricial dimension at most $2$.