Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

Construction of Gr\"obner bases. S-Polynomials and Standard Representations


Christoph Schwarzweller
University of T\"ubingen

Summary.

We continue the Mizar formalization of Gr\"{o}bner bases following [6]. In this article we introduce S-polynomials and standard representations and show how these notions can be used to characterize Gr\"{o}bner bases.

MML Identifier: GROEB_2

The terminology and notation used in this paper have been introduced in the following articles [23] [31] [32] [34] [33] [8] [3] [15] [28] [30] [9] [7] [5] [14] [12] [19] [18] [24] [27] [17] [1] [4] [13] [21] [20] [29] [26] [16] [10] [25] [2] [22] [11] [35]

Contents (PDF format)

  1. Preliminaries
  2. S-Polynomials
  3. Standard Representations

Bibliography

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Journal of Formalized Mathematics, 12, 2000.
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[4] Grzegorz Bancerek. Reduction relations. Journal of Formalized Mathematics, 7, 1995.
[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[6] Thomas Becker and Volker Weispfenning. \em Gr\"obner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.
[7] Jozef Bialas. Group and field definitions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[9] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[10] Agata Darmochwal and Yatsuka Nakamura. The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[11] Jaroslaw Kotowicz. Functions and finite sequences of real numbers. Journal of Formalized Mathematics, 5, 1993.
[12] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[13] Gilbert Lee and Piotr Rudnicki. On ordering of bags. Journal of Formalized Mathematics, 14, 2002.
[14] Michal Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Journal of Formalized Mathematics, 2, 1990.
[15] Michal Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Journal of Formalized Mathematics, 2, 1990.
[16] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Journal of Formalized Mathematics, 5, 1993.
[17] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Journal of Formalized Mathematics, 11, 1999.
[18] Christoph Schwarzweller. The binomial theorem for algebraic structures. Journal of Formalized Mathematics, 12, 2000.
[19] Christoph Schwarzweller. More on multivariate polynomials: Monomials and constant polynomials. Journal of Formalized Mathematics, 13, 2001.
[20] Christoph Schwarzweller. Polynomial reduction. Journal of Formalized Mathematics, 14, 2002.
[21] Christoph Schwarzweller. Term orders. Journal of Formalized Mathematics, 14, 2002.
[22] Christoph Schwarzweller. Characterization and existence of Gr\"obner bases. Journal of Formalized Mathematics, 15, 2003.
[23] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[24] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[25] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[26] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[27] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.
[28] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[29] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[30] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[31] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[32] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[33] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[34] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.
[35] Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Journal of Formalized Mathematics, 12, 2000.

Received June 11, 2003


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