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GRÖBNER-SHIRSHOV BASIS FOR MONOMIALS SEMIRING OVER D-A RINGS
Keywords:
multivariete polynomials, divisible and annihilable ring, semi-ring, Gröbner-Shirshov basis, overlap relationsAbstract
The study of Gröbner basis over a D-A ring has been introduced by Kapur and Cai [2], where the set of monomials is a commutative monoid, and an algorithm for computing commutative Gröbner bases has been provided. In 2013, Bokut et al. [7] proposed a Gröbner-Shirshov algorithm over a field where the set of monomials (commutative or not) is a semiring. The work of Bokut et al. was generalized in [17] over valuation ring in 2020.
In this paper, we generalize these two methods and propose a Gröbner-Shirshov method over a D-A ring where the set of monomials (commutative or not) is a semiring.
Received: January 7, 2023
Accepted: February 10, 2023
References
B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal, Ph. D. Thesis, University of Innsbruck, 1965.
Deepak Kapur and Yongyang Cai, An algorithm for computing a Gröbner basis of a polynomial ideal over a ring avec zero divisors, Mathematics in Computer Science 2 (2009), 601-634.
Nafissatou Diarra, Noncommutative Gröbner Bases over D-A rings, Ph. D Thesis, Universite Cheikh Anta Diop de Dakar-Bibliotheque Universitair, 2017.
L. A. Bokut, Imbedding into simple associative algebras, Algebra Logic 15 (1976), 117-142.
L. A. Bokut, Yuqun Chen and Qiuhui Mo, Gröbner-Shirshov bases and embeddings of algebras, Internat. J. Algebra Comput. 20 (2010), 875-900.
L. A. Bokut, Yuqun Chen and Jianjun Qiu, Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras, J. Pure Appl. Algebra 214 (2010), 89-100.
L. A. Bokut, Y. Chen and Q. Mo, Gröbner-Shirshov bases for semirings, Journal of Algebra 385 (2013), 47-63.
D. Kapur and Y. Cai, An algorithm for computing a Gröbner basis of a polynomial ideal over a ring with zero divisors, Math. Comput. Sci. 2 (2009), 601-634.
D. Kapur and A. Kandri-Rody, Computing a Gröbner basis for a polynomial ideal over a Euclidean domain, J. Symbol. Comput. 6(1) (1988), 37-57.
I. Yengui, Dynamical Gröbner basis, J. Algebra 301(2) (2006), 447-458.
A. S. E. M. Bouesso and D. Sow, Noncommutative Gröbner bases over rings, Comm. Algebra 43(2) (2015), 541-557.
X. Xiu, Non-commutative Gröbner bases and applications, Ph. D. Thesis, Department of Informatics and Mathematics, University of Passau, 2012.
A. I. Shirshov, Some algorithmic problem for Lie algebras, Sibirsk. Mat. Z. 3 (1962), 292-296 (in Russian); English translation in SIGSAM Bull. 33(2) (1999), 3-6.
M. Fiore and T. Leinster, An objective representation of the Gaussian integers, J. Symbol. Comput. 37 (2004), 707-716.
A. Blass, Seven trees in one, J. Pure Appl. Algebra 103 (1995), 1-21.
G. M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218.
Y. Diop, L. Mesmoudi and D. Sow, Semi-ring based Gröbner-Shirshov over a Noetherian valuation ring, Associative and Non-associative Algebras and Applications, Springer Proceedings in Mathematics and Statistics, Springer, Cham, Vol. 311, 2020, pp. 183-198.
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