DOI QR코드

DOI QR Code

A CHARACTERIZATION OF FINITE FACTORIZATION POSITIVE MONOIDS

  • Polo, Harold (Mathematics Department University of Florida)
  • 투고 : 2021.08.07
  • 심사 : 2021.10.29
  • 발행 : 2022.07.31

초록

We provide a characterization of the positive monoids (i.e., additive submonoids of the nonnegative real numbers) that satisfy the finite factorization property. As a result, we establish that positive monoids with well-ordered generating sets satisfy the finite factorization property, while positive monoids with co-well-ordered generating sets satisfy this property if and only if they satisfy the bounded factorization property.

키워드

과제정보

I am grateful to Felix Gotti for his guidance during the preparation of this paper, in particular, for many useful conversations that lead up to the discovery of Theorem 3.3. While working on the same, I was generously supported by the University of Florida Mathematics Department Fellowship and the CAM Summer Research Fellowship.

참고문헌

  1. D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), no. 1, 1-19. https://doi.org/10.1016/0022-4049(90)90074-R
  2. D. D. Anderson and B. Mullins, Finite factorization domains, Proc. Amer. Math. Soc. 124 (1996), no. 2, 389-396. https://doi.org/10.1090/S0002-9939-96-03284-4
  3. N. R. Baeth, S. T. Chapman, and F. Gotti, Bi-atomic classes of positive semirings, Semigroup Forum 103 (2021), no. 1, 1-23. https://doi.org/10.1007/s00233-021-10189-8
  4. M. Bras-Amor'os, Increasingly enumerable submonoids of R: music theory as a unifying theme, Amer. Math. Monthly 127 (2020), no. 1, 33-44. https://doi.org/10.1080/00029890.2020.1674073
  5. P. Cesarz, S. T. Chapman, S. McAdam, and G. J. Schaeffer, Elastic properties of some semirings defined by positive systems, in Commutative algebra and its applications, 89-101, Walter de Gruyter, Berlin,2009.
  6. S. T. Chapman, F. Gotti, and M. Gotti, When is a Puiseux monoid atomic?, Amer. Math. Monthly 128 (2021), no. 4, 302-321. https://doi.org/10.1080/00029890.2021.1865064
  7. P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251-264. https://doi.org/10.1017/s0305004100042791
  8. J. Correa-Morris and F. Gotti, On the additive structure of algebraic valuations cyclic free semirings, Preprint on arXiv: https://arxiv.org/pdf/2008.13073.pdf
  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. 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
  11. A. Geroldinger, F. Gotti, and S. Tringali, On strongly primary monoids, with a focus on Puiseux monoids, J. Algebra 567 (2021), 310-345. https://doi.org/10.1016/j.jalgebra.2020.09.019
  12. 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
  13. R. Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.
  14. 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
  15. F. Gotti and M. Gotti, Atomicity and boundedness of monotone Puiseux monoids, Semigroup Forum 96 (2018), no. 3, 536-552. https://doi.org/10.1007/s00233-017-9899-9
  16. 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
  17. F. Halter-Koch, Finiteness theorems for factorizations, Semigroup Forum 44 (1992), no. 1, 112-117. https://doi.org/10.1007/BF02574329
  18. B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202-252. https://doi.org/10.2307/1990552
  19. H. Polo, On the sets of lengths of Puiseux monoids generated by multiple geometric sequences, Commun. Korean Math. Soc. 35 (2020), no. 4, 1057-1073. https://doi.org/10.4134/CKMS.c200017
  20. J. C. Rosales and P. A. Garcia-Sanchez, Numerical semigroups, Developments in Mathematics, 20, Springer, New York, 2009. https://doi.org/10.1007/978-1-4419-0160-6