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Blanco, Víctor; García-Sánchez, Pedro A.; Geroldinger, Alfred

Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids. (English) Zbl 1474.20117

Actes Rencontres C.I.R.M. 2, No. 2, 95-98 (2010).
Summary: Arithmetical invariants – such as sets of lengths, catenary and tame degrees – describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

Cited in 1 Document

MSC:

20M14 Commutative semigroups
20M13 Arithmetic theory of semigroups
13A05 Divisibility and factorizations in commutative rings

Keywords:

presentations for semigroups; catenary degree; tame degree; sets of lengths; numerical monoid; Krull monoid

Software:

numericalsgps
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Full Text: DOI arXiv

References:

[1] J. Amos, S.T. Chapman, N. Hine, and J. Paixao, \(Sets of lengths do not characterize numerical monoids\), Integers 7 (2007), Paper A50, 8p. · Zbl 1139.20056
[2] V. Blanco, P. A. García-Sánchez, and A. Geroldinger, \(Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids\), manuscript. · Zbl 1279.20072
[3] S.T. Chapman, P.A. García-Sánchez, and D. Llena, \(The catenary and tame degree of numerical monoids\), Forum Math. 21 (2009), 117 - 129. · Zbl 1177.20070
[4] S.T. Chapman, P.A. García-Sánchez, D. Llena, V. Ponomarenko, and J.C. Rosales, \(The catenary and tame degree in finitely generated commutative cancellative monoids\), Manuscr. Math. 120 (2006), 253 - 264. · Zbl 1117.20045
[5] M. Delgado, P.A. García-Sánchez, and J. Morais, \(``numericalsgps'': a \)gap\( package on numerical semigroups\), ().
[6] M. Freeze and A. Geroldinger, \(Unions of sets of lengths\), Funct. Approximatio, Comment. Math. 39 (2008), 149 - 162. · Zbl 1228.20046
[7] W. Gao and A. Geroldinger, \(On products of \)k\( atoms\), Monatsh. Math. 156 (2009), 141 - 157. · Zbl 1184.20051
[8] P.A. García-Sánchez and I. Ojeda, \(Uniquely presented finitely generated commutative monoids\), Pacific J. Math. 248 (2010), 91 - 105. · Zbl 1208.20052
[9] A. Geroldinger, \(Additive group theory and non-unique factorizations\), Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Ruzsa, eds.), Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, 2009, pp. 1 - 86. · Zbl 1221.20045
[10] A. Geroldinger and F. Halter-Koch, \(Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory\), Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006. · Zbl 1113.11002
[11] A. Geroldinger and F. Kainrath, \(On the arithmetic of tame monoids with applications to Krull monoids and Mori domains\), J. Pure Appl. Algebra 214 (2010), 2199 - 2218. · Zbl 1207.20055
[12] F. Kainrath, \(Arithmetic of Mori domains and monoids\) :\( the Global Case\), manuscript. · Zbl 1394.20035
[13] A. Philipp, \(A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings\), Semigroup Forum. · Zbl 1213.20059
[14] —, \(A precise result on the arithmetic of non-principal orders in algebraic number fields\), manuscript. · Zbl 1303.11126
[15] —, \(A characterization of arithmetical invariants by the monoid of relations\), Semigroup Forum 81 (2010), 424 - 434. · Zbl 1213.20059
[16] J.C. Rosales and P.A. García-Sánchez, \(Finitely Generated Commutative Monoids\), Nova Science Publishers, 1999. · Zbl 0966.20028
[17] W.A. Schmid, \(A realization theorem for sets of lengths\), J. Number Theory 129 (2009), 990 - 999. | Copyright Cellule MathDoc 2018 · Zbl 1191.11031
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