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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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ISSN
1898-9934
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Volume 22, Issue 1

Issues

Definition of Flat Poset and Existence Theorems for Recursive Call

Kazuhisa Ishida / Yasunari Shidama / Adam Grabowski
Published Online: 2014-03-30 | DOI: https://doi.org/10.2478/forma-2014-0001

Summary

This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set [12] in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like BaseFunc or RecFunc. The remaining part builds the background for information engineering approach for lists, namely recursive call for posets [21].We formalized some facts from Chapter 8 of this book as an introduction to the next two sections where we concentrate on binary product of posets rather than on a more general case.

Keywords: flat posets; recursive calls for posets; flattening operator

References

  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

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

  • [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [5] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.Google Scholar

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

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

  • [8] Czesław Bylinski. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.Google Scholar

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

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

  • [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [12] B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.Google Scholar

  • [13] Marek Dudzicz. Representation theorem for finite distributive lattices. Formalized Mathematics, 9(2):261-264, 2001.Google Scholar

  • [14] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501-505, 1996.Web of ScienceGoogle Scholar

  • [15] Kazuhisa Ishida and Yasunari Shidama. Fixpoint theorem for continuous functions on chain-complete posets. Formalized Mathematics, 18(1):47-51, 2010. doi:10.2478/v10037-010-0006-x.CrossrefGoogle Scholar

  • [16] Artur Korniłowicz. Cartesian products of relations and relational structures. Formalized Mathematics, 6(1):145-152, 1997.Google Scholar

  • [17] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Google Scholar

  • [18] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Google Scholar

  • [19] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.Google Scholar

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

  • [21] Glynn Winskel. The Formal Semantics of Programming Languages. The MIT Press, 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] Mariusz Zynel and Czesław Bylinski. Properties of relational structures, posets, lattices and maps. Formalized Mathematics, 6(1):123-130, 1997.Google Scholar

About the article

Published Online: 2014-03-30


Citation Information: Formalized Mathematics, Volume 22, Issue 1, Pages 1–10, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2014-0001.

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© by Kazuhisa Ishida. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0) BY-SA 3.0

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