대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제35권4호
  • /
  • Pages.1057-1073
  • /
  • 2020
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE SETS OF LENGTHS OF PUISEUX MONOIDS GENERATED BY MULTIPLE GEOMETRIC SEQUENCES

  • Polo, Harold (Department of Mathematics University of Florida)
  • 투고 : 2020.01.18
  • 심사 : 2020.07.20
  • 발행 : 2020.10.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study some of the factorization aspects of rational multicyclic monoids, that is, additive submonoids of the nonnegative rational numbers generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids M that are hereditarily atomic (i.e., every submonoid of M is atomic). Additionally, we show that the sets of lengths of certain rational multicyclic monoids are finite unions of multidimensional arithmetic progressions, while their unions satisfy the Structure Theorem for Unions of Sets of Lengths. Finally, we realize arithmetic progressions as the sets of distances of some additive submonoids of the nonnegative rational numbers.

키워드

과제정보

The author wants to thank Felix Gotti for his guidance throughout the different stages of this manuscript and for many useful conversations about factorization theory. The author extends his thanks to anonymous referees whose feedback improve the final version of this paper. While working on this manuscript, the author was supported by the University of Florida Mathematics Department Fellowship.

참고문헌

  1. D. Bachman, N. R. Baeth, and J. Gossell, Factorizations of upper triangular matrices, Linear Algebra Appl. 450 (2014), 138-157. https://doi.org/10.1016/j.laa.2014.02.038
  2. N. R. Baeth and A. Geroldinger, Monoids of modules and arithmetic of direct-sum decompositions, Pacific J. Math. 271 (2014), no. 2, 257-319. https://doi.org/10.2140/pjm.2014.271.257
  3. C. Bowles, S. T. Chapman, N. Kaplan, and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006), no. 5, 695-718. https://doi.org/10.1142/S0219498806001958
  4. S. T. Chapman, J. Daigle, R. Hoyer, and N. Kaplan, Delta sets of numerical monoids using nonminimal sets of generators, Comm. Algebra 38 (2010), no. 7, 2622-2634. https://doi.org/10.1080/00927870903045165
  5. S. T. Chapman, F. Gotti, and M. Gotti, Factorization invariants of Puiseux monoids generated by geometric sequences, Comm. Algebra 48 (2020), no. 1, 380-396. https://doi.org/10.1080/00927872.2019.1646269
  6. S. T. Chapman, F. Gotti, and M. Gotti, When is a Puiseux monoid atomic?, To appear in Amer. Math. Monthly.
  7. S. T. Chapman and W. W. Smith, Factorization in Dedekind domains with finite class group, Israel J. Math. 71 (1990), no. 1, 65-95. https://doi.org/10.1007/BF02807251
  8. S. Colton and N. Kaplan, The realization problem for delta sets of numerical semigroups, J. Commut. Algebra 9 (2017), no. 3, 313-339. https://doi.org/10.1216/JCA-2017-9-3-313
  9. J. Coykendall and F. Gotti, On the atomicity of monoid algebras, J. Algebra 539 (2019), 138-151. https://doi.org/10.1016/j.jalgebra.2019.07.033
  10. Y. Fan, A. Geroldinger, F. Kainrath, and S. Tringali, Arithmetic of commutative semi-groups with a focus on semigroups of ideals and modules, J. Algebra Appl. 16 (2017), no. 12, 1750234, 42 pp. https://doi.org/10.1142/S0219498817502346
  11. A. Foroutan and W. Hassler, Factorization of powers in C-monoids, J. Algebra 304 (2006), no. 2, 755-781. https://doi.org/10.1016/j.jalgebra.2005.11.006
  12. M. Freeze and A. Geroldinger, Unions of sets of lengths, Funct. Approx. Comment. Math. 39 (2008), part 1, 149-162. https://doi.org/10.7169/facm/1229696561
  13. W. Gao and A. Geroldinger, On products of k atoms, Monatsh. Math. 156 (2009), no. 2, 141-157. https://doi.org/10.1007/s00605-008-0547-z
  14. P. A. Garcia-Sanchez, D. Llena, and A. Moscariello, Delta sets for symmetric numerical semigroups with embedding dimension three, Aequationes Math. 91 (2017), no. 3, 579-600. https://doi.org/10.1007/s00010-017-0474-y
  15. A. Geroldinger, Sets of lengths, Amer. Math. Monthly 123 (2016), no. 10, 960-988. https://doi.org/10.4169/amer.math.monthly.123.10.960
  16. A. Geroldinger, F. Gotti, and S. Tringali, On strongly primary monoids, with a focus on Puiseux monoids, preprint.
  17. A. Geroldinger and F. Halter-Koch, Non-unique factorizations, Pure and Applied Mathematics (Boca Raton), 278, Chapman & Hall/CRC, Boca Raton, FL, 2006. https://doi.org/10.1201/9781420003208
  18. A. Geroldinger and W. A. Schmid, A realization theorem for sets of distances, J. Algebra 481 (2017), 188-198. https://doi.org/10.1016/j.jalgebra.2017.03.003
  19. A. Geroldinger, W. A. Schmid, and Q. Zhong, Systems of sets of lengths: transfer Krull monoids versus weakly Krull monoids, in Rings, polynomials, and modules, 191-235, Springer, Cham, 2017.
  20. A. Geroldinger and Q. Zhong, Sets of arithmetical invariants in transfer Krull monoids, J. Pure Appl. Algebra 223 (2019), no. 9, 3889-3918. https://doi.org/10.1016/j.jpaa.2018.12.011
  21. R. Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.
  22. F. Gotti, On the atomic structure of Puiseux monoids, J. Algebra Appl. 16 (2017), no. 7, 1750126, 20 pp. https://doi.org/10.1142/S0219498817501262
  23. F. Gotti, Puiseux monoids and transfer homomorphisms, J. Algebra 516 (2018), 95-114. https://doi.org/10.1016/j.jalgebra.2018.08.026
  24. F. Gotti, Systems of sets of lengths of Puiseux monoids, J. Pure Appl. Algebra 223 (2019), no. 5, 1856-1868. https://doi.org/10.1016/j.jpaa.2018.08.004
  25. F. Gotti, Increasing positive monoids of ordered fields are FF-monoids, J. Algebra 518 (2019), 40-56. https://doi.org/10.1016/j.jalgebra.2018.10.010
  26. F. Gotti, The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid, J. Algebra Appl. https://doi.org/10.1142/S0219498820501376
  27. F. Gotti and M. Gotti, Atomicity and boundedness of monotone Puiseux monoids, Semi-group Forum 96 (2018), no. 3, 536-552. https://doi.org/10.1007/s00233-017-9899-9
  28. F. Gotti and C. O'Neill, The elasticity of Puiseux monoids, To appear in J. Commut. Algebra. doi:https://projecteuclid.org/euclid.jca/1523433696
  29. A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Cambridge Philos. Soc. 75 (1974), 321-329. https://doi.org/10.1017/s0305004100048532
  30. F. Halter-Koch, Finitely generated monoids, finitely primary monoids, and factorization properties of integral domains, in Factorization in integral domains (Iowa City, IA, 1996), 31-72, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997.
  31. D. Smertnig, Sets of lengths in maximal orders in central simple algebras, J. Algebra 390 (2013), 1-43. https://doi.org/10.1016/j.jalgebra.2013.05.016
  32. S. Tringali, Structural properties of subadditive families with applications to factorization theory, Israel J. Math. 234 (2019), no. 1, 1-35. https://doi.org/10.1007/s11856-019-1922-2