Principal ideals of finitely generated commutative monoids. (English) Zbl 1003.20052

Summary: We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.


20M14 Commutative semigroups
20M12 Ideal theory for semigroups
20M05 Free semigroups, generators and relations, word problems
Full Text: DOI EuDML


[1] T. Becker and W. Weispfenning: Gröbner Bases: a Computational Approach to Commutative Algebra. Springer-Verlag, New York, 1993. · Zbl 0772.13010
[2] A. H. Clifford: The Algebraic Theory of Semigroups. Amer. Math. Soc., Providence, 1961. · Zbl 0111.03403
[3] D. Eisenbud and B. Sturmfels: Binomial ideals. Duke Math. J. 84 (1996), 1-45. · Zbl 0873.13021
[4] R. Gilmer: Commutative Semigroup Rings. University of Chicago Press, Chicago, 1984. · Zbl 0566.20050
[5] J. Herzog: Generators and relations of abelian semigroup and semigroups rings. Manuscripta Math. 3 (1970), 175-193. · Zbl 0211.33801
[6] G. B. Preston: Rédei’s characterization of congruences of finitely generated free commutative semigroups. Acta Math. Acad. Sci. Hungar. 26 (1975), 337-342. · Zbl 0338.20070
[7] L. Rédei: The theory of finitely commutative semigroups. Pergamon, Oxford-Edinburgh-New York, 1965.
[8] J. C. Rosales and P. A. García-Sánchez: Finitely generated commutative monoids. vol. xiv, Nova Science Publishers, New York, 1999. · Zbl 0966.20028
[9] J. C. Rosales and J. M. Urbano-Blanco: A deterministic algorithm to decide if a finitely presented monoid is cancellative. Comm. Algebra 24 (1996), 4217-4224. · Zbl 0940.20058
[10] J. C. Rosales: On finitely generated submonoids of \(N^k\). Semigroup Forum 50 (1995), 251-262. · Zbl 0831.20080
[11] J. C. Rosales and P. A. García-Sánchez: Presentations for subsemigroups of finitely generated commutative semigroups. Israel J. Math. 113 (1999), 269-283. · Zbl 0943.20061
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