Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

**Piotr Rudnicki**- University of Alberta, Edmonton
- This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

- This article completes the Mizar formalization of Chapter I, Section 2 from [13]. After presenting some preliminary material (not all of which is later used in this article) we give the proof of theorem 2.7 (i), p.60. We do not follow the hint from [13] suggesting using the equations 2.3, p. 58. The proof is taken directly from the definition of continuous lattice. The goal of the last section is to prove the correspondence between the set of all congruences of a continuous lattice and the set of all kernel operators of the lattice which preserve directed sups (Corollary 2.13).

- Preliminaries
- Some Remarks on Lattice Product
- Kernel Projections and Quotient Lattices

- [1]
Grzegorz Bancerek.
Curried and uncurried functions.
*Journal of Formalized Mathematics*, 2, 1990. - [2]
Grzegorz Bancerek.
Complete lattices.
*Journal of Formalized Mathematics*, 4, 1992. - [3]
Grzegorz Bancerek.
Bounds in posets and relational substructures.
*Journal of Formalized Mathematics*, 8, 1996. - [4]
Grzegorz Bancerek.
Directed sets, nets, ideals, filters, and maps.
*Journal of Formalized Mathematics*, 8, 1996. - [5]
Grzegorz Bancerek.
The ``way-below'' relation.
*Journal of Formalized Mathematics*, 8, 1996. - [6]
Czeslaw Bylinski.
Basic functions and operations on functions.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Czeslaw Bylinski.
Galois connections.
*Journal of Formalized Mathematics*, 8, 1996. - [12]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [13] 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.
- [14]
Adam Grabowski and Robert Milewski.
Boolean posets, posets under inclusion and products of relational structures.
*Journal of Formalized Mathematics*, 8, 1996. - [15]
Artur Kornilowicz.
Cartesian products of relations and relational structures.
*Journal of Formalized Mathematics*, 8, 1996. - [16]
Robert Milewski.
Completely-irreducible elements.
*Journal of Formalized Mathematics*, 10, 1998. - [17]
Beata Padlewska.
Families of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Konrad Raczkowski and Pawel Sadowski.
Equivalence relations and classes of abstraction.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Andrzej Trybulec.
Domains and their Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [22]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [23]
Wojciech A. Trybulec.
Partially ordered sets.
*Journal of Formalized Mathematics*, 1, 1989. - [24]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [25]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Edmund Woronowicz and Anna Zalewska.
Properties of binary relations.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Mariusz Zynel and Czeslaw Bylinski.
Properties of relational structures, posets, lattices and maps.
*Journal of Formalized Mathematics*, 8, 1996.

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