Abstract
A numerical semigroup is a submonoid of \({\mathbb {Z}}_{\ge 0}\) for which its complement in \({\mathbb {Z}}_{\ge 0}\) is finite. For any set of positive integers a, b, c, the numerical semigroup S(a, b, c) formed by the set of solutions of the inequality \(ax \bmod {b} \le cx\) is said to be proportionally modular. For any interval \([\alpha ,\beta ]\), \(S\big ([\alpha ,\beta ]\big )\) is the submonoid of \({\mathbb {Z}}_{\ge 0}\) obtained by intersecting the submonoid of \({\mathbb {Q}}_{\ge 0}\) generated by \([\alpha ,\beta ]\) with \({\mathbb {Z}}_{\ge 0}\). For the numerical semigroup S of embedding dimension 3, we characterize a, b, c and \(\alpha ,\beta \) such that both S(a, b, c) and \(S\big ([\alpha ,\beta ]\big )\) equal S.
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The authors are grateful to the referee for pointing out a minor error in one of the proofs, and for some general suggestions.
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Elizeche, E.F., Tripathi, A. On proportionally modular numerical semigroups of embedding dimension three. Ramanujan J 61, 1197–1211 (2023). https://doi.org/10.1007/s11139-022-00680-3
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DOI: https://doi.org/10.1007/s11139-022-00680-3