Representation Theorem for Free Continuous Lattices

Journal of Formalized Mathematics
Volume 10, 1998
University of Bialystok
Copyright (c) 1998 Association of Mizar Users

Representation Theorem for Free Continuous Lattices

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

Summary.

We present the Mizar formalization of theorem 4.17, Chapter I from [13]: a free continuous lattice with $m$ generators is isomorphic to the lattice of filters of $2^X$ ($\overline{\overline{X}} = m$) which is freely generated by $\{\uparrow x : x \in X\}$ (the set of ultrafilters).

MML Identifier: WAYBEL22

The terminology and notation used in this paper have been introduced in the following articles [20] [10] [25] [18] [26] [8] [9] [3] [12] [16] [1] [2] [19] [24] [4] [22] [23] [17] [21] [5] [14] [27] [6] [11] [7] [15]

Contents (PDF format)

Preliminaries
Free Generators of Continuous Lattices
Representation Theorem for Free Continuous Lattices

Bibliography

[1] Grzegorz Bancerek. Cardinal 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] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[9] Czeslaw Bylinski. Functions from a set to a set. 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] Mariusz Giero. More on products of many sorted algebras. Journal of Formalized Mathematics, 8, 1996.

[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] Robert Milewski. Completely-irreducible elements. Journal of Formalized Mathematics, 10, 1998.

[16] Michal Muzalewski. Categories of groups. Journal of Formalized Mathematics, 3, 1991.

[17] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Journal of Formalized Mathematics, 4, 1992.

[18] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.

[19] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.

[20] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[21] Andrzej Trybulec. Function domains and Fr\aenkel operator. Journal of Formalized Mathematics, 2, 1990.

[22] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.

[23] Andrzej Trybulec. Many sorted algebras. Journal of Formalized Mathematics, 6, 1994.

[24] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.

[25] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[26] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[27] Mariusz Zynel and Czeslaw Bylinski. Properties of relational structures, posets, lattices and maps. Journal of Formalized Mathematics, 8, 1996.

Received July 21, 1998

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