Positioned numerical semigroups with maximal gender as function of multiplicity and Frobenius number

Abstract

A $C$-semigroup (respectively a $D$-semigroup) is a positioned numerical semigroup $S$ such that $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-1}{2}$ (respectively $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-2}{2}$). In this paper we study these semigroups giving formulas for the Frobenius number, pseudo-Frobenius number, and type. Furthermore, we give algorithms for computing the whole set of $C$-semigroups and $D$-semigroups.

Keywords

• numerical semigroups
• positioned numerical semigroups
• $C$-semigroups
• $D$-semigroups tree
• Frobenius number
• multiplicity and gender

References

1. [1] V. Barucci, D.E. Dobbs and M. Fontana, Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains, Memoirs of the Amer. Math. Soc. 598, 1997.
2. [2] M.B. Branco, M.C. Faria and J.C. Rosales, Positioned numerical semigroups, Quaest. Math. 44, 679-691, 2021.
3. [3] M.B. Branco, M.C. Faria and J.C. Rosales, Almost-positioned numerical semigroups, Results Math. 76, 1-14, 2021.
4. [4] M.B. Branco, I. Ojeda and J.C. Rosales, The set of numerical semigroups of a given multiplicity and Frobenius number, Port. Math. 78, 147-167, 2021.
5. [5] R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum 35, 63-83, 1987.
6. [6] E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25, 748-751, 1973.
7. [7] J.C. Rosales, On symmetric numerical semigroups, J. Algebra 182, 422-434, 1996.
8. [8] J.C. Rosales, Adding or removing on element from a Pseudo- symmetric numerical semigroup, Boll. Unione Mat. Ital. 9, 681-696, 2006.

9. [9] J.C. Rosales, Numerical semigroups that differ from a Symmetric numerical semigroups in one element, Algebra Colloq. 15, 23-32, 2008.
10. [10] J.C. Rosales and M.B. Branco, Irreducible numerical semigroups, Pacific J. Math. 209, 131-143, 2003.
11. [11] J.C. Rosales and P.A. García-Sánchez, Numerical semigroups, Springer Science & Business Media, 2009.