Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

Closure Operators and Subalgebras


Grzegorz Bancerek
Warsaw University, Bialystok

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

MML Identifier: WAYBEL10

The terminology and notation used in this paper have been introduced in the following articles [13] [8] [15] [16] [17] [5] [7] [6] [1] [12] [14] [2] [10] [3] [4] [11] [18] [9]

Contents (PDF format)

  1. Preliminaries
  2. The lattice of closure operators
  3. The lattice of closure systems
  4. Isomorphism between closure operators and closure systems
  5. Isomorphism between closure operators preserving directed sups and subalgebras

Bibliography

[1] Grzegorz Bancerek. K\"onig's theorem. 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] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Galois connections. Journal of Formalized Mathematics, 8, 1996.
[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] Beata Madras. Product of family of universal algebras. Journal of Formalized Mathematics, 5, 1993.
[13] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[14] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.
[15] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[16] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[17] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[18] Mariusz Zynel and Czeslaw Bylinski. Properties of relational structures, posets, lattices and maps. Journal of Formalized Mathematics, 8, 1996.

Received January 15, 1997

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