On Delta Sets Of Some Pseudo-Symmetric Numerical Semigroups With Embedding Dimension Three

Let S be a numerical semigroup. The catenary degree of an element s in S is a non-negative integer used to measure the distance between factorizations of s. The catenary degree of the numerical semigroup S is obtained at the maximum catenary degree of its elements. The maximum catenary degree of S is attained via Betti elements of S with complex properties. The Betti elements of S can be obtained from all minimal presentations of S. A presentation for S is a system of generators of the kernel congruence of the special factorization homomorphism. A presentation is minimal if it can not be converted to another presentation, that is, any of its proper subsets is no longer a presentation. The Delta set of S is a factorization invariant measuring the complexity of sets of the factorization lengths for the elements in S. In this study, we will mainly express the given above invariants of a special pseudo-symmetric numerical semigroup family in terms of its generators.

PDF

___

[1] A. Assi and P.A. García-Sánchez, Numerical semigroups and applications. Springer, Cham: RSME Springer Series, 2016.
[2] S. T. Chapman, P. A. García-Sánchez, Z. Tripp and C. Viola, C. “Measuring primality in numerical semigroups with embedding dimension three,” Journal of Algebra and Its Applications, vol. 15, no. 1, pp. 16, 2016.
[3] S. T. Chapman, R. Hoyer and N. Kaplan, “Delta Sets of Numerical Monoids are Eventually Periodic,” Aequationes Math., vol. 77, pp. 273-279, 2009.
[4] S. T. Chapman, N. Kaplan, J. Daigle and R. Hoyer, “Delta Sets of Numerical Monoids Using NonMinimal Sets of Generators,” Comm. Algebra, vol. 38, pp. 2622-2634, 2010.
[5] S. T. Chapman, N. Kaplan, T. Lemburg, A. Niles and C. Zlogar, “Shifts of Generators and Delta Sets of Numerical Monoids,” Internat. J. Algebra Comp., vol. 24, no. 5, pp. 655–669, 2014.
[6] R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Williams and B. Wissman, “Minimal presentations of shifted numerical monoids,” International Journal of Algebra and Computation, vol. 28, no. 1, pp. 53–68, 2018.
[7] R. Conaway, M. Williams, J. Horton and F. Gotti, “Shifting numerical semigroups” Allen Institute for Artificial Intelligence. 2015. [Online]. Available: https://www.semanticscholar.org. [Accessed: Dec. 12, 2020].
[8] M. Delgado, P. A. García-Sánchez and J. Morais, ““numericalsgps”: a gap package on numerical semigroups,” 2020. [Online]. Available: https://www.gap-system.org/Packages/numericalsgps.html. [Accessed: Nov. 11, 2021].
[9] P.A. García-Sánchez, D. Llena and A. Moscariello, “Delta sets for numerical semigroups with embedding dimension three,” 2015. [Online]. Available: https://arxiv.org/abs/1504.02116v1 [Accessed: Nov. 11, 2021].
[10] A. Geroldinger, “On the arithmetic of certain not integrally closed Noetherian integral domains,” Comm. Algebra, vol. 19, pp. 685–698, 1991.
[11] A. Geroldinger and F. Halter-Koch, Non-unique factorizations: Algebraic, combinatorial and analytic theory, Boca Raton-London-New York: Chapman and Hall/CRC, 2006.
[12] S. İlhan and M. Süer, “On a class of pseudo symmetric numerical semigroups,” JP Journal of Algebra, Number Theory and Applications, vol. 20, no. 2, pp. 225-230, 2011.
[13] S. İlhan and M. Süer, “Gaps of a class of pseudo symmetric numerical semigroups,” Acta Universitestis Apulensis, vol. 34, pp. 99-104, 2013.
[14] D. Narsingh, Graph Theory with Applications to Engineering and Computer Science. The United States of America, USA: Prentice Hall Series in Automatic Computation, 1974.
[15] C. O’Neil, V. Ponomorenko, R. Tate and G. Webb, “On the set of catenary degrees of finitely generated cancellative commutative monoids,” International Journal of Algebra and Computation, vol. 26, no. 3, pp. 565-576, 2016.
[16] J. C. Rosales and M. B. Branco, “Irreducible numerical semigroups with arbitrary multiplicity and embedding dimension,” J. Algebra, vol. 264, pp. 305–315, 2003.
[17] J. C. Rosales and P. A. García-Sánchez, Finitely generated commutative monoids. New York: Nova 28 Science Publishers, 1999.
[18] J. C. Rosales and P. A. García-Sánchez, Numerical semigroups. In: Developments in Mathematics. New York: Springer, 2009.
[19] M. S. Schwartz, “Factorization Lengths in Numerical Monoids,” 2019. [Online]. Available: https://digitalcommons.bard.edu/cgi/viewcontent.cgi?article=1034&context=senproj_s2019 [Accessed: Nov. 11, 2021].