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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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1898-9934
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Volume 21, Issue 3 (Oct 2013)

Issues

Prime Filters and Ideals in Distributive Lattices

Adam Grabowski

Summary.

The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone representation theorem [18] for Boolean or Heyting lattices. The main aim of the present article was to bridge this gap between general distributive lattices and Boolean algebras, having in mind that the more general approach will eventually replace the common proof of aforementioned articles.1

Because in Boolean algebras the notions of ultrafilters, prime filters and maximal filters coincide, we decided to construct some concrete examples of ultrafilters in nontrivial Boolean lattice. We proved also the Prime Ideal Theorem not as BPI (Boolean Prime Ideal), but in the more general setting.

In the final section we present Nachbin theorems [15],[1] expressed both in terms of maximal and prime filters and as the unordered spectra of a lattice [11], [10]. This shows that if the notion of maximal and prime filters coincide in the lattice, it is Boolean.

Keywords: prime filters; prime ideals; distributive lattices

  • [1] Raymond Balbes and Philip Dwinger. Distributive Lattices. University of Missouri Press, 1975.Google Scholar

  • [2] Grzegorz Bancerek. Filters - part I. Formalized Mathematics, 1(5):813-819, 1990.Google Scholar

  • [3] Grzegorz Bancerek. Ideals. Formalized Mathematics, 5(2):149-156, 1996.Google Scholar

  • [4] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719-725, 1991.Google Scholar

  • [5] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [6] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [7] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [9] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.Google Scholar

  • [10] George Grätzer. General Lattice Theory. Academic Press, New York, 1978.Google Scholar

  • [11] George Grätzer. Lattice Theory: Foundation. Birkhäuser, 2011.Google Scholar

  • [12] Jolanta Kamienska. Representation theorem for Heyting lattices. Formalized Mathematics, 4(1):41-45, 1993.Google Scholar

  • [13] Jolanta Kamienska and Jarosław Stanisław Walijewski. Homomorphisms of lattices, finite join and finite meet. Formalized Mathematics, 4(1):35-40, 1993.Google Scholar

  • [14] Agnieszka Julia Marasik. Boolean properties of lattices. Formalized Mathematics, 5(1): 31-35, 1996.Google Scholar

  • [15] Leopoldo Nachbin. Une propriété characteristique des algebres booleiennes. Portugaliae Mathematica, 6:115-118, 1947.Google Scholar

  • [16] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Google Scholar

  • [17] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Google Scholar

  • [18] Marshall H. Stone. The theory of representations of Boolean algebras. Transactions of the American Mathematical Society, 40:37-111, 1936.Google Scholar

  • [19] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.Google Scholar

  • [20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [21] Jarosław Stanisław Walijewski. Representation theorem for Boolean algebras. Formalized Mathematics, 4(1):45-50, 1993.Google Scholar

  • [22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Google Scholar

  • [23] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Google Scholar

  • [24] Stanisław Zukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215-222, 1990. Google Scholar

About the article

Published in Print: 2013-10-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0023.

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[1]
Adam Grabowski
Journal of Automated Reasoning, 2015, Volume 55, Number 3, Page 211

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