Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

**Beata Madras**

- In the paper open and order generating subsets are defined. Irreducible and prime elements are also defined. The article includes definitions and facts presented in [16, pp. 68-72].

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

- Preliminaries
- Open sets
- Irreducible elements
- Order generating sets
- Prime elements

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [3]
Grzegorz Bancerek.
Cartesian product of functions.
*Journal of Formalized Mathematics*, 3, 1991. - [4]
Grzegorz Bancerek.
Complete lattices.
*Journal of Formalized Mathematics*, 4, 1992. - [5]
Grzegorz Bancerek.
Bounds in posets and relational substructures.
*Journal of Formalized Mathematics*, 8, 1996. - [6]
Grzegorz Bancerek.
Directed sets, nets, ideals, filters, and maps.
*Journal of Formalized Mathematics*, 8, 1996. - [7]
Grzegorz Bancerek.
The ``way-below'' relation.
*Journal of Formalized Mathematics*, 8, 1996. - [8]
Grzegorz Bancerek and Andrzej Trybulec.
Miscellaneous facts about functions.
*Journal of Formalized Mathematics*, 8, 1996. - [9]
Jozef Bialas.
Group and field definitions.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Czeslaw Bylinski.
Basic functions and operations on functions.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Czeslaw Bylinski.
Galois connections.
*Journal of Formalized Mathematics*, 8, 1996. - [15]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [16] 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.
- [17]
Adam Grabowski.
Auxiliary and approximating relations.
*Journal of Formalized Mathematics*, 8, 1996. - [18]
Adam Grabowski and Robert Milewski.
Boolean posets, posets under inclusion and products of relational structures.
*Journal of Formalized Mathematics*, 8, 1996. - [19]
Artur Kornilowicz.
Meet -- continuous lattices.
*Journal of Formalized Mathematics*, 8, 1996. - [20]
Beata Madras.
Product of family of universal algebras.
*Journal of Formalized Mathematics*, 5, 1993. - [21]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [23]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [24]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [25]
Wojciech A. Trybulec.
Partially ordered sets.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Edmund Woronowicz and Anna Zalewska.
Properties of binary relations.
*Journal of Formalized Mathematics*, 1, 1989. - [29]
Mariusz Zynel.
The equational characterization of continuous lattices.
*Journal of Formalized Mathematics*, 8, 1996. - [30]
Mariusz Zynel and Czeslaw Bylinski.
Properties of relational structures, posets, lattices and maps.
*Journal of Formalized Mathematics*, 8, 1996.

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