Journal of Formalized Mathematics
Volume 14, 2002
University of Bialystok
Copyright (c) 2002 Association of Mizar Users

On Ordering of Bags


Gilbert Lee
University of Alberta, Edmonton
Piotr Rudnicki
University of Alberta, Edmonton

Summary.

We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings are not necessarily well orderings.

MML Identifier: BAGORDER

The terminology and notation used in this paper have been introduced in the following articles [33] [12] [41] [42] [44] [43] [16] [37] [3] [34] [4] [38] [40] [29] [36] [19] [2] [9] [21] [6] [1] [28] [35] [24] [5] [23] [27] [7] [20] [30] [15] [26] [18] [17] [11] [10] [14] [25] [31] [39] [32] [13] [22]

Contents (PDF format)

  1. Preliminaries
  2. More About Bags
  3. Some Special Orders
  4. Ordering of Finite Subsets

Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[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. The well ordering relations. Journal of Formalized Mathematics, 1, 1989.
[5] Grzegorz Bancerek. K\"onig's theorem. Journal of Formalized Mathematics, 2, 1990.
[6] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Journal of Formalized Mathematics, 8, 1996.
[7] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[8] Thomas Becker and Volker Weispfenning. \em Gr\"obner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.
[9] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[10] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[11] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[12] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[13] Czeslaw Bylinski. Binary operations applied to finite sequences. Journal of Formalized Mathematics, 2, 1990.
[14] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.
[15] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
[16] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[17] Adam Grabowski. Auxiliary and approximating relations. Journal of Formalized Mathematics, 8, 1996.
[18] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Journal of Formalized Mathematics, 8, 1996.
[19] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[20] Andrzej Kondracki. The Chinese Remainder Theorem. Journal of Formalized Mathematics, 9, 1997.
[21] Jaroslaw Kotowicz. Monotone real sequences. Subsequences. Journal of Formalized Mathematics, 1, 1989.
[22] Jaroslaw Kotowicz. Functions and finite sequences of real numbers. Journal of Formalized Mathematics, 5, 1993.
[23] Gilbert Lee and Piotr Rudnicki. Dickson's lemma. Journal of Formalized Mathematics, 14, 2002.
[24] Beata Madras. Product of family of universal algebras. Journal of Formalized Mathematics, 5, 1993.
[25] Beata Madras. On the concept of the triangulation. Journal of Formalized Mathematics, 7, 1995.
[26] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Journal of Formalized Mathematics, 6, 1994.
[27] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Journal of Formalized Mathematics, 5, 1993.
[28] Jan Popiolek. Real normed space. Journal of Formalized Mathematics, 2, 1990.
[29] Piotr Rudnicki and Andrzej Trybulec. On same equivalents of well-foundedness. Journal of Formalized Mathematics, 9, 1997.
[30] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Journal of Formalized Mathematics, 11, 1999.
[31] Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Journal of Formalized Mathematics, 12, 2000.
[32] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[33] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[34] Andrzej Trybulec. Tuples, projections and Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[35] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[36] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[37] Andrzej Trybulec and Agata Darmochwal. Boolean domains. Journal of Formalized Mathematics, 1, 1989.
[38] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.
[39] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[40] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Journal of Formalized Mathematics, 1, 1989.
[41] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[42] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[43] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[44] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.

Received March 12, 2002

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