On the internal sum of Puiseux monoids
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Abstract
In this paper, we investigate the internal (finite) sum of submonoids of rank-
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References
[1] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorizations in integral domains, J. Pure Appl. Algebra 69 (1990) 1–19. https://doi.org/10.1016/0022-4049(90)90074-R Google Scholar
[2] D. F. Anderson and F. Gotti, Bounded and finite factorization domains. In: Rings, Monoids, and Module Theory (Eds. A. Badawi and J. Coykendall), Springer Proceedings in Mathematics & Statistics, vol. 382, Singapore, 2022, pp. 7–57. http://dx.doi.org/10.1007/978-981-16-8422-7_2 Google Scholar
[3] S. T. Chapman, F. Gotti, and M. Gotti, When is a Puiseux monoid atomic?, Amer. Math. Monthly 128 (2021) 302–321. http://dx.doi.org/10.1080/00029890.2021.1865064 Google Scholar
[4] P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968) 251–264. https://doi.org/10.1017/S0305004100042791 Google Scholar
[5] J. Coykendall and F. Gotti, Atomicity in integral domains. In: Recent Progress in Rings and Factorization Theory, Springer, 2025. http://dx.doi.org/10.48550/arXiv.2406.02503 Google Scholar
[6] L. Fuchs, Infinite Abelian Groups I, Academic Press, 1970. Google Scholar
[7] A. Geroldinger, Sets of lengths, Amer. Math. Monthly 123 (2016) 960–988. http://dx.doi.org/10.4169/amer.math.monthly.123.10.960 Google Scholar
[8] A. Geroldinger and F. Gotti, On monoid algebras having every nonempty subset of ( mathbb{N}_{geq 2} ) as a length set. Submitted. Preprint on arXiv: https://arxiv.org/pdf/2404.11494 Google Scholar
[9] 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 Google Scholar
[10] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics Vol. 278, Chapman & Hall/CRC, Boca Raton, 2006. http://dx.doi.org/10.1201/9781420003208 Google Scholar
[11] A. Geroldinger and Q. Zhong, A characterization of length-factorial Krull monoids, New York J. Math. 27 (2021) 1347–1374. http://dx.doi.org/10.48550/arXiv.2101.10908 Google Scholar
[12] A. Geroldinger and Q. Zhong, Factorization theory in commutative monoids, Semigroup Forum 100 (2020) 22–51. https://doi.org/10.1007/s00233-019-10079-0 Google Scholar
[13] R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Mathematics, The University of Chicago Press, London, 1984. Google Scholar
[14] F. Gotti, Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid, Linear Algebra Appl. 604 (2020) 146–186. http://dx.doi.org/10.1016/j.laa.2020.06.009 Google Scholar
[15] F. Gotti, Puiseux monoids and transfer homomorphisms, J. Algebra 516 (2018) 95–114. https://doi.org/10.1016/J.JALGEBRA.2018.08.026 Google Scholar
[16] 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 Google Scholar
[17] A. Grams, Atomic rings and the ascending chain condition for principal ideals. Math. Proc. Cambridge Philos. Soc. 75 (1974) 321–329. Google Scholar
[18] F. Halter-Koch, Finiteness theorems for factorizations, Semigroup Forum 44 (1992) 112–117. Google Scholar
[19] F. Halter-Koch, Ideal Systems: An Introduction to Multiplicative Ideal Theory, Marcel Dekker, 1998. Google Scholar