On the sets of lengths of Puiseux monoids generated by multiple geometric sequences
2
조회
0
다운로드
1
인용
0
참고문헌
키워드
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.
Abstract
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.
참고문헌 (32)
- D. Bachman, "Factorizations of upper triangular matrices", Linear Algebra Appl, 450, pp.138~157 (2014)[DOI]
- N. R. Baeth, "Monoids of modules and arithmetic of direct-sum decompositions", Pacific J. Math, 271(2), pp.257~319 (2014)[DOI]
- C. Bowles, "On delta sets of numerical monoids", J. Algebra Appl, 5(5), pp.695~718 (2006)[DOI]
- S. T. Chapman, "Delta sets of numerical monoids using nonminimal sets of generators", Comm. Algebra, 38(7), pp.2622~2634 (2010)[DOI]
- S. T. Chapman, "Factorization invariants of Puiseux monoids generated by geometric sequences", Comm. Algebra, 48(1), pp.380~396 (2020)[DOI]
- S. T. Chapman, "When is a Puiseux monoid atomic?", Amer. Math. Monthly
- S. T. Chapman, "Factorization in Dedekind domains with finite class group", Israel J. Math, 71(1), pp.65~95 (1990)[DOI]
- S. Colton, "The realization problem for delta sets of numerical semigroups", J. Commut. Algebra, 9(3), pp.313~339 (2017)[DOI]
- J. Coykendall, "On the atomicity of monoid algebras", J. Algebra, 539, pp.138~151 (2019)[DOI]
- Y. Fan, "Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules", J. Algebra Appl, 16(12), pp.1750234~1750275 (2017)[DOI]
- A. Foroutan, "Factorization of powers in C-monoids", J. Algebra, 304(2), pp.755~781 (2006)[DOI]
- M. Freeze, "Unions of sets of lengths", Funct. Approx. Comment. Math., 39(part 1), pp.149~162 (2008)[DOI]
- W. Gao, "On products of k atoms", Monatsh. Math, 156(2), pp.141~157 (2009)[DOI]
- P. A. Garc´ıa-S´anchez, "Delta sets for symmetric numerical semigroups with embedding dimension three", Aequationes Math, 91(3), pp.579~600 (2017)[DOI]
- A. Geroldinger, "Sets of lengths", Amer. Math. Monthly, 123(10), pp.960~988 (2016)[DOI]
- A. Geroldinger, "On strongly primary monoids, with a focus on Puiseux monoids", ~
- A. Geroldinger, "Non-unique factorizations", Pure and Applied Mathematics (Boca Raton), 278 (2006), Chapman & Hall[DOI]
- A. Geroldinger, "A realization theorem for sets of distances", J. Algebra, 481, pp.188~198 (2017)[DOI]
- A. Geroldinger, "Rings, polynomials, and modules", pp.191~235 (2017), Springer
- A. Geroldinger, "Sets of arithmetical invariants in transfer Krull monoids", J. Pure Appl. Algebra, 223(9), pp.3889~3918 (2019)[DOI]
- R. Gilmer, "Commutative semigroup rings, Chicago Lectures in Mathematics" (1984), University of Chicago Press
- F. Gotti, "On the atomic structure of Puiseux monoids", J. Algebra Appl, 16(7), pp.1750126~1750145 (2017)[DOI]
- F. Gotti, "Puiseux monoids and transfer homomorphisms", J. Algebra, 516, pp.95~114 (2018)[DOI]
- F. Gotti, "Systems of sets of lengths of Puiseux monoids", J. Pure Appl. Algebra, 223(5), pp.1856~1868 (2019)[DOI]
- F. Gotti, "Increasing positive monoids of ordered fields are FF-monoids", J. Algebra, 518, pp.40~56 (2019)[DOI]
- F. Gotti, "The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid", J. Algebra Appl[DOI]
- F. Gotti, "Atomicity and boundedness of monotone Puiseux monoids", Semigroup Forum, 96(3), pp.536~552 (2018)[DOI]
- F. Gotti, "The elasticity of Puiseux monoids", J. Commut. Algebra
- A. Grams, "Atomic rings and the ascending chain condition for principal ideals", Proc. Cambridge Philos. Soc, 75, pp.321~329 (1974)[DOI]
- F. Halter-Koch, "Finitely generated monoids, finitely primary monoids, and factorization properties of integral domains, in Factorization in integral domains (Iowa City, IA, 1996)", Lecture Notes in Pure and Appl. Math, 189, pp.31~72 (1996), Dekker
- D. Smertnig, "Sets of lengths in maximal orders in central simple algebras", J. Algebra, 390, pp.1~43 (2013)[DOI]
- S. Tringali, "Structural properties of subadditive families with applications to factorization theory", Israel J. Math, 234(1), pp.1~35 (2019)[DOI]