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.

MML Identifier: WAYBEL_6

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]

Contents (PDF format)

  1. Preliminaries
  2. Open sets
  3. Irreducible elements
  4. Order generating sets
  5. 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


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