×

One-generated semirings and additive divisibility. (English) Zbl 1358.16038

Semirings \((S,+,\cdot)\) in the sense of this paper are commutative with respect to both operations and have an identity \(1\). Such semirings are called additively divisible, if for any \(a\in S\) and every non-negative integer \(n\) there is some \(b\in S\) such that \(a=b+\ldots + b\) (\(n\)-times). It is an open conjecture that every finitely generated additively divisible semiring is additively idempotent. By sophisticated rewriting techniques, similar as in the Buchberger algorithm in the semiring \({\mathbb N}[x]\) of nonzero polynomials over the non-negative integers, the authors prove this conjecture for semirings generated by one single element. As a second result, it is shown that every at most countable commutative semigroup \((A,+)\) can be embedded in the additive semigroup of some one-generated semiring \((S,+,\cdot)\) as above.

MSC:

16Y60 Semirings
12Y05 Computational aspects of field theory and polynomials (MSC2010)
20M14 Commutative semigroups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bokut, L. A., Chen, Y. and Mo, Q., Gröbner-Shirshov bases for semirings, J. Algebra385 (2013) 47-63. · Zbl 1286.16038
[2] Clark, W. E., Holland, W. C. and Székely, G. J., Decompositions in discrete semigroups, Studia Sc. Math. Hungarica34 (1998) 15-23. · Zbl 0930.20049
[3] Delgado, M., Rosales, J. C. and García-Sánchez, P. A., Numerical semigroups problem list, CIM Bull.33 (2013) 15-26.
[4] El Bashir, R., Hurt, J., Jančařík, A. and Kepka, T., Simple commutative semirings, J. Algebra236 (2001) 277-306. · Zbl 0976.16034
[5] P. A. García-Sánchez, Numerical semigroups mini-course, http://www.ugr.es/\( \sim\) pedro/minicurso-porto.pdf.
[6] Golan, J. S., Semirings and Their Applications (Kluwer Academic Publishers, Dordrecht, 1999). · Zbl 0947.16034
[7] Golan, J. S., Semirings and Affine Equations Over Them: Theory and Applications (Kluwer Academic Publishers, Dordrecht, 2003). · Zbl 1042.16038
[8] Gunawardena, J., An introduction to idempotency, in Idempotency, ed. Gunawardena, J. (Cambridge University Press, Cambridge, 1998), pp. 1-49. · Zbl 0898.16032
[9] Hebisch, U. and Weinert, H. J., Semirings: Algebraic Theory and Applications in Computer Science (World Scientific Publishing, Singapore, 1998). · Zbl 0934.16046
[10] Ježek, J., Kala, V. and Kepka, T., Finitely generated algebraic structures with various divisibility conditions, Forum Math.24(2) (2012) 379-397. · Zbl 1254.16041
[11] Kala, V. and Kepka, T., A note on finitely generated ideal-simple commutative semirings, Comment. Math. Univ. Carolin.49(1) (2008) 1-9. · Zbl 1192.16045
[12] Kala, V., Kepka, T. and Korbelář, M., Notes on commutative parasemifields, Comment. Math. Univ. Carolin.50(4) (2009) 521-533. · Zbl 1203.16038
[13] T. Kepka and M. Korbelár, Conjectures on additively divisible commutative semirings, to appear in Math. Slovaca. · Zbl 1399.16118
[14] Otto, F. and Sokratova, O., Reduction relations for monoid semirings, J. Symbolic Comput.37 (2004) 343-376. · Zbl 1121.68350
[15] Rosales, J. C. and García-Sánchez, P. A., Numerical Semigroups, , Vol. 20 (Springer, 2009). · Zbl 1220.20047
[16] Tamura, T., Minimal commutative divisible semigroups, Bull. Amer. Math. Soc.69 (1963) 713-716. · Zbl 0123.01801
[17] Zumbrägel, J., Classification of finite congruence-simple semirings with zero, J. Algebra Appl.7(4) (2008) 363-377. · Zbl 1155.16036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.