Journal of Formalized Mathematics
Volume 1, 1989
University of Bialystok
Copyright (c) 1989 Association of Mizar Users

Partially Ordered Sets


Wojciech A. Trybulec
Warsaw University
Supported by RPBP.III-24.C1.

Summary.

In the beginning of this article we define the choice function of a non-empty set family that does not contain $\emptyset$ as introduced in [6, pages 88-89]. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure non-empty set and order of the set, chains, lower and upper cone of a subset, initial segments of element and subset of partially ordered set. Some theorems that belong rather to [5] or [12] are proved.

MML Identifier: ORDERS_1

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

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. The well ordering relations. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[6] Kazimierz Kuratowski. \em Wstep do teorii mnogosci i topologii. PWN, War\-sza\-wa, 1977.
[7] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[8] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[9] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[10] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[11] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[12] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.

Received August 30, 1989


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