Acknowledgement
We are grateful to our mentors, Prof. Jim Coykendall and Dr. Felix Gotti, for proposing this project and for their guidance during the preparation of this project. While working on this paper, we were part of PRIMES-USA, a year-long math research program hosted by MIT. We would like to express our collective gratitude to the MIT PRIMES program for arranging such an engaging research experience.
References
- K. Ajran, J. Bringas, B. Li, E. Singer, and M. Tirador, Factorization in additive monoids of evaluation polynomial semirings, Comm. Algebra 51 (2023), no. 10, 4347-4362. https://doi.org/10.1080/00927872.2023.2208672
- 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
- D. F. Anderson and F. Gotti, Bounded and finite factorization domains, in Rings, Monoids and Module Theory, 7-57, Springer Proc. Math. Stat., 382, Springer, Singapore. https://doi.org/10.1007/978-981-16-8422-7_2
- N. R. Baeth, S. T. Chapman, and F. Gotti, Bi-atomic classes of positive semirings, Semi-group Forum 103 (2021), no. 1, 1-23. https://doi.org/10.1007/s00233-021-10189-8
- N. R. Baeth and F. Gotti, Factorizations in upper triangular matrices over information semialgebras, J. Algebra 562 (2020), 466-496. https://doi.org/10.1016/j.jalgebra.2020.06.031
- A. Baker, Transcendental Number Theory, second edition, Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 1990.
- 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
- M. Bras-Amor'os and M. Gotti, Atomicity and density of Puiseux monoids, Comm. Algebra 49 (2021), no. 4, 1560-1570. https://doi.org/10.1080/00927872.2020.1840574
- A. Bu, J. Vulakh, and A. Zhao, Length-factoriality and pure irreducibility, Commun. Algebra 51 (2023), no. 9, 3745-3755. https://doi.org/10.1080/00927872.2023.2187629
- S. T. Chapman and J. Coykendall, Half-factorial domains, a survey, in Non-Noetherian Commutative Ring Theory, 97-115, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, Springer, Boston 2020. https://doi.org/10.1007/978-1-4757-3180-4_5
- S. T. Chapman, J. Coykendall, F. Gotti, and W. W. Smith, Length-factoriality in commutative monoids and integral domains, J. Algebra 578 (2021), 186-212. https://doi.org/10.1016/j.jalgebra.2021.03.010
- S. T. Chapman and M. Gotti, Atomicity of positive monoids, submitted.
- 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
- P. M. Cohn, Bezout rings and their subrings, Math. Proc. Cambridge Philos. Soc. 64 (1968), no. 2, 251-264. https://doi.org/10.1017/s0305004100042791
- J. Correa-Morris and F. Gotti, On the additive structure of algebraic valuations of polynomial semirings, J. Pure Appl. Algebra 226 (2022), no. 11, Paper No. 107104, 20 pp. https://doi.org/10.1016/j.jpaa.2022.107104
- J. Coykendall, D. E. Dobbs, and B. Mullins, On integral domains with no atoms, Comm. Algebra 27 (1999), no. 12, 5813-5831. https://doi.org/10.1080/00927879908826792
- J. Coykendall and W. W. Smith, On unique factorization domains, J. Algebra 332 (2011), no. 1, 62-70. https://doi.org/10.1016/j.jalgebra.2010.10.024
- W. Gao, C. Liu, S. Tringali, and Q. Zhong, On half-factoriality of transfer Krull monoids, Comm. Algebra 49 (2021), no. 1, 409-420. https://doi.org/10.1080/00927872.2020.1800720
- A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations, Pure and Applied Mathematics, 278, Chapman & Hall/CRC, Boca Raton, FL, 2006. https://doi.org/10.1201/9781420003208
- A. Geroldinger and Q. Zhong, A characterization of length-factorial Krull monoids, New York J. Math. 27 (2021), 1347-1374.
- J. S. Golan, Semirings and their Applications, Kluwer Acad. Publ., Dordrecht, 1999. https://doi.org/10.1007/978-94-015-9333-5
- 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
- F. Gotti, On semigroup algebras with rational exponents, Comm. Algebra 50 (2022), no. 1, 3-18. https://doi.org/10.1080/00927872.2021.1949018
- 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
- F. Gotti and M. Gotti, On the molecules of numerical semigroups, Puiseux monoids, and Puiseux algebras, in Numerical Semigroups, 141-161, Springer INdAM Ser., 40, Springer, Cham, 2020. https://doi.org/10.1007/978-3-030-40822-0_10
- F. Gotti and B. Li, Atomic semigroup rings and the ascending chain condition on principal ideals, Proc. Amer. Math. Soc. 151 (2023), no. 6, 2291-2302. https://doi.org/10.1090/proc/16295
- F. Gotti and H. Polo, On the arithmetic of polynomial semidomains, Forum Math. 35 (2023), no. 5, 1179-1197. https://doi.org/10.1515/forum-2022-0091
- F. Gotti and H. Polo, On the subatomicity of polynomial semidomains, in Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, 197-212, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-28847-0_13
- F. Gotti and J. Vulakh, On the atomic structure of torsion-free monoids, Semigroup Forum 107 (2023), no. 2, 402-423. https://doi.org/10.1007/s00233-023-10385-8
- 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
- F. Halter-Koch, Finiteness theorems for factorizations, Semigroup Forum 44 (1992), no. 1, 112-117. https://doi.org/10.1007/BF02574329
- N. Jiang, B. Li, and S. Zhu, On the primality and elasticity of algebraic valuations of cyclic free semirings, Internat. J. Algebra Comput. 33 (2023), no. 2, 197-210. https://doi.org/10.1142/S021819672350011X
- H. Polo, A characterization of finite factorization positive monoids, Commun. Korean Math. Soc. 37 (2022), no. 3, 669-679. https://doi.org/10.4134/CKMS.c210270
- A. Zaks, Half factorial domains, Bull. Amer. Math. Soc. 82 (1976), no. 5, 721-723. https://doi.org/10.1090/S0002-9904-1976-14130-4
- S. Zhu, Factorizations in evaluation monoids of Laurent semirings, Comm. Algebra 50 (2022), no. 6, 2719-2730. https://doi.org/10.1080/00927872.2021.2018449