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Frobenius Allowable Gaps of Generalized Numerical Semigroups

Singhal, Deepesh and Lin, Yuxin (2022) Frobenius Allowable Gaps of Generalized Numerical Semigroups. Electronic Journal of Combinatorics, 29 (4). Art. No. P4.12. ISSN 1077-8926. doi:10.37236/10748.

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A generalized numerical semigroup is a submonoid S of ℕᵈ for which the complement ℕᵈ\S is finite. The points in the complement ℕᵈ\S are called gaps. A gap F is considered Frobenius allowable if there is some relaxed monomial ordering on ℕᵈ with respect to which F is the largest gap. We characterize the Frobenius allowable gaps of a generalized numerical semigroup. A generalized numerical semigroup that has only one maximal gap under the natural partial ordering of ℕᵈ is called a Frobenius generalized numerical semigroup. We show that Frobenius generalized numerical semigroups are precisely those whose Frobenius gap does not depend on the relaxed monomial ordering. We estimate the number of Frobenius generalized numerical semigroup with a given Frobenius gap F = (F^(1), . . . , F^(d)) ∈ ℕᵈ and show that it is close to √3^[(F^(1) + 1) . . . (F^(d) + 1)] for large d. We define notions of quasi-irreducibility and quasi-symmetry for generalized numerical semigroups. While in the case of d = 1 these notions coincide with irreducibility and symmetry, they are distinct in higher dimensions.

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Lin, Yuxin0000-0001-9230-7728
Additional Information:We would like to thank Nathan Kaplan for several helpful discussions.
Issue or Number:4
Record Number:CaltechAUTHORS:20221115-642560900.6
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117882
Deposited By: Research Services Depository
Deposited On:29 Nov 2022 16:33
Last Modified:29 Nov 2022 17:24

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