No Access

On the set of catenary degrees of finitely generated cancellative commutative monoids

Christopher O’Neill

Mathematics Department, Texas A&M University, College Station, TX 77843, United States

E-mail Address: coneill@math.tamu.edu

Corresponding author.

Search for more papers by this author

Vadim Ponomarenko

Mathematics Department, San Diego State University, San Diego, CA 92182, United States

E-mail Address: vponomarenko@mail.sdsu.edu

Search for more papers by this author

Reuben Tate

Mathematics Department, University of Hawai‘i at Hilo, Hilo, HI 96720, United States

E-mail Address: reubent@hawaii.edu

Search for more papers by this author

, and

Gautam Webb

Mathematics Department, Colorado College, Colorado Springs, CO 80903, United States

E-mail Address: gautam.webb@coloradocollege.edu

Search for more papers by this author

https://doi.org/10.1142/S0218196716500247Cited by:8 (Source: Crossref)

Abstract

The catenary degree of an element $n$ of a cancellative commutative monoid $S$ is a nonnegative integer measuring the distance between the irreducible factorizations of $n$ . The catenary degree of the monoid $S$ , defined as the supremum over all catenary degrees occurring in $S$ , has been studied as an invariant of nonunique factorization. In this paper, we investigate the set $C (S)$ of catenary degrees achieved by elements of $S$ , focusing on the case where $S$ is finitely generated (where $C (S)$ is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of $C (S)$ that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for $C (S)$ .

Communicated by B. Steinberg

Keywords:

References

1. D. F. Anderson and S. T. Chapman, How far is an element from being prime? J. Algebra Appl. 9 (2010) 1–11. Link, Web of Science, Google Scholar
2. V. Blanco, P. García-Sánchez and A. Geroldinger, Semigroup-theoretical characterizations of arithmetical invariants with applications to numberical monoids and Krull monoids, Illinois J. Math. 55(4) (2011) 1385–1414. Google Scholar
3. C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24. Link, Web of Science, Google Scholar
4. S. T. Chapman, M. Corrales, A. Miller, C. Miller and D. Phatel, The catenary and tame degrees on a numerical monoid are eventually periodic, J. Aust. Math. Soc. 97(3) (2014) 289–300. Web of Science, Google Scholar
5. S. T. Chapman, M. Corrales, A. Miller, C. Miller and D. Phatel, The catenary degrees of elements in numerical monoids generated by arithmetic sequences, accepted, J. Algebra Appl. Google Scholar
6. S. Chapman, P. García-Sánchez and D. Llena, The catenary and tame degree of numerical semigroups, Forum Math. 21 (2009) 117–129. Web of Science, Google Scholar
7. S. Chapman, P. García-Sánchez, D. Llena, V. Ponomarenko and J. Rosales, The catenary and tame degree in finitely generated commutative cancellative monoids, Manuscripta Math. 120(3) (2006) 253–264. Web of Science, Google Scholar
8. S. T. Chapman, M. T. Holden and T. A. Moore, Full elasticity in atomic monoids and integral domains, Rocky Mountain J. Math. 36 (2006) 1437–1455. Web of Science, Google Scholar
9. S. Colton and N. Kaplan, The realization problem for delta sets of numerical semigroups, preprint. Available at arXiv: math.AC/1503.08496. Google Scholar
10. M. Delgado, P. García-Sánchez and J. Morais, NumericalSgps, A package for numerical semigroups, Version 0.980 dev (2013), (GAP package), http://www.fc.up.pt/cmup/mdelgado/numericalsgps/. Google Scholar
11. P. García-Sánchez, D. Llena and A. Moscariello, Delta sets for numerical semigroups with embedding dimension three, preprint. Available at arXiv: math.AC/1504.02116. Google Scholar
12. P. García Sánchez, I. Ojeda and J. Rosales, Affine semigroups having a unique Betti element, J. Algebra Appl. 12(3) (2013) 1250177, 11 pp. Web of Science, Google Scholar
13. P. A. García-Sánchez, I. Ojeda and R. Sánchez and A. Navarro, Factorization invariants in half-factorial affine semigroups, Internat. J. Algebra Comput. 23(1) (2013) 111–122. Link, Web of Science, Google Scholar
14. P. A. García-Sánchez and J. C. Rosales, Numerical Semigroups, Vol. 20 (Developments in Mathematics, Springer-Verlag, 2009). Google Scholar
15. A. Geroldinger, D. J. Grynkiewicz and W. A. Schmid, The catenary degree of Krull monoids I, J. Théor. Nombres Bordx. 23 (2011) 137–169. Web of Science, Google Scholar
16. A. Geroldinger and F. Halter-Koch, Nonunique factorization, Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Vol. 278 (Chapman & Hall/CRC, 2006). Google Scholar
17. A. Geroldinger and Q. Zhong, The catenary degree of Krull monoids II, J. Australian Math. Soc. 98 (2015) 324–354. Web of Science, Google Scholar
18. C. O’Neill and R. Pelayo, On the linearity of $ω$ -primality in numerical monoids, J. Pure and Applied Algebra 218 (2014) 1620–1627. Web of Science, Google Scholar
19. M. Omidali, The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences, Forum Mathematicum 24(3) (2012) 627–640. Web of Science, Google Scholar
20. A. Philipp, A characterization of arithmetical invariants by the monoid of relations, Semigroup Forum 81(3) (2010) 424–434. Web of Science, Google Scholar
21. V. Ponomarenko, Minimal zero sequences of finite cyclic groups, Integers 4(A24) (2004) 6 pp. Google Scholar
22. Sage: Open Source Mathematics Software, available at www.sagemath.org. Google Scholar