Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001 Association of Mizar Users

Hierarchies and Classifications of Sets


Mariusz Giero
University of Bialystok

Summary.

This article is a continuation of [3] article. Further properties of classification of sets are proved. The notion of hierarchy of a set is introduced. Properties of partitions and hierarchies are shown. The main theorem says that for each hierarchy there exists a classification which union equals to the considered hierarchy.

This work has been partially supported by the European Community TYPES grant IST-1999-29001 and CALCULEMUS grant HPRN-CT-2000-00102.

MML Identifier: TAXONOM2

The terminology and notation used in this paper have been introduced in the following articles [9] [2] [12] [6] [13] [1] [14] [10] [7] [8] [5] [11] [4] [3]

Contents (PDF format)

  1. Tree and Classification of a Set
  2. The Hierarchy of a Set
  3. Some Properties of Partitions, Hierarchies and Classifications of Sets

Acknowledgments

I would like to thank Prof. Andrzej Trybulec for his help in the preparation of this article.

Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[3] Mariusz Giero and Roman Matuszewski. Lower tolerance. Preliminaries to Wroclaw taxonomy. Journal of Formalized Mathematics, 12, 2000.
[4] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Journal of Formalized Mathematics, 8, 1996.
[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Journal of Formalized Mathematics, 10, 1998.
[6] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[7] Konrad Raczkowski and Pawel Sadowski. Equivalence relations and classes of abstraction. Journal of Formalized Mathematics, 1, 1989.
[8] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Journal of Formalized Mathematics, 9, 1997.
[9] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[10] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.
[11] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Journal of Formalized Mathematics, 1, 1989.
[12] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[13] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[14] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.

Received December 28, 2001

[ Download a postscript version, MML identifier index, Mizar home page]