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

Chains on a Grating in Euclidean Space


Freek Wiedijk
University of Nijmegen

Summary.

Translation of pages 101, the second half of 102, and 103 of [16].

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

MML Identifier: CHAIN_1

The terminology and notation used in this paper have been introduced in the following articles [20] [10] [23] [24] [3] [8] [13] [9] [18] [1] [19] [15] [4] [7] [14] [17] [2] [21] [12] [5] [11] [22] [6]

Contents (PDF format)

  1. Preliminaries
  2. Gratings, Cells, Chains, Cycles
  3. Groups and Homomorphisms

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. K\"onig's theorem. Journal of Formalized Mathematics, 2, 1990.
[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[6] Jozef Bialas. Group and field definitions. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.
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[12] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[13] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[14] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[15] Michal Muzalewski. Opposite rings, modules and their morphisms. Journal of Formalized Mathematics, 4, 1992.
[16] M. H. A. Newman. \em Elements of the Topology of Plane Sets of Points. Cambridge University Press, 1951.
[17] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[18] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Journal of Formalized Mathematics, 9, 1997.
[19] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[20] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[21] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[22] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[23] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[24] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received January 27, 2003

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