Irreducible and Prime Elements
Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996
Association of Mizar Users
Irreducible and Prime Elements
-
Beata Madras
-
Summary.
-
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.
The terminology and notation used in this paper have been
introduced in the following articles
[22]
[13]
[26]
[24]
[15]
[27]
[1]
[28]
[9]
[25]
[21]
[2]
[4]
[11]
[12]
[10]
[3]
[23]
[20]
[5]
[18]
[6]
[14]
[30]
[19]
[7]
[17]
[29]
[8]
-
Preliminaries
-
Open sets
-
Irreducible elements
-
Order generating sets
-
Prime elements
Bibliography
- [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.
Received December 1, 1996
[
Download a postscript version,
MML identifier index,
Mizar home page]