Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## Properties of Relational Structures, Posets, Lattices and Maps

Mariusz Zynel
Warsaw University, Bialystok
Czeslaw Bylinski
Warsaw University, Bialystok

### Summary.

In the paper we present some auxiliary facts concerning posets and maps between them. Our main purpose, however is to give an account on complete lattices and lattices of ideals. A sufficient condition that a lattice might be complete, the fixed-point theorem and two remarks upon images of complete lattices in monotone maps, introduced in [9, pp. 8-9], can be found in Section~7. Section~8 deals with lattices of ideals. We examine the meet and join of two ideals. In order to show that the lattice of ideals is complete, the infinite intersection of ideals is investigated.

This work was partially supported by Office of Naval Research Grant N00014-95-1-1336.

#### MML Identifier: YELLOW_2

The terminology and notation used in this paper have been introduced in the following articles [15] [8] [17] [18] [6] [7] [13] [2] [1] [16] [14] [3] [10] [4] [11] [5] [12]

#### Contents (PDF format)

1. Basic Facts
2. Relational Substructures
3. Maps
4. The Image of a Map
5. Monotone Maps
6. Idempotent Maps
7. Complete Lattices
8. Lattices of Ideals
9. Special Maps
10. The Family of Elements in a Lattice

#### Bibliography

[1] Grzegorz Bancerek. The well ordering relations. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. Complete lattices. Journal of Formalized Mathematics, 4, 1992.
[3] Grzegorz Bancerek. Quantales. Journal of Formalized Mathematics, 6, 1994.
[4] Grzegorz Bancerek. Bounds in posets and relational substructures. Journal of Formalized Mathematics, 8, 1996.
[5] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Journal of Formalized Mathematics, 8, 1996.
[6] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. \em A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.
[10] Adam Grabowski. On the category of posets. Journal of Formalized Mathematics, 8, 1996.
[11] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Journal of Formalized Mathematics, 8, 1996.
[12] Michal Muzalewski. Categories of groups. Journal of Formalized Mathematics, 3, 1991.
[13] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[14] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[16] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.
[17] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[18] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received September 20, 1996