On the genus of a quotient of a numerical semigroup
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We find a relation between the genus of a quotient of a numerical semigroup S and the genus of S itself. We use this identity to compute the genus of a quotient of S when S has embedding dimension 2. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.
KeywordsNumerical semigroups Quotient of numerical semigroups Genus Frobenius numbers
A. Adeniran, C. Defant, Y. Gao, C. Hettle, Q. Liang, H. Nam, and A. Volk were partially supported by NSF-DMS Grant #1603823 “Collaborative Research: Rocky Mountain Great Plains Graduate Research Workshops in Combinatorics” and by NSF-DMS Grant #1604458, “Collaborative Research: Rocky Mountain Great Plains Graduate Research Workshops in Combinatorics.” S. Butler and P. E. Harris were partially supported by NSA Grant #H98230-18-1-0017, “The 2018 and 2019 Rocky Mountain—Great Plains Graduate Research Workshops in Combinatorics.” C. Defant was also supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Research Fellowship.
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