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]
-
Preliminaries
-
S-Polynomials
-
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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