Formalization of Trellises and Tolerance Relations

Adam Grabowski; Franciszek Turowski

doi:10.2478/forma-2024-0022

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Formalization of Trellises and Tolerance Relations

Formalized Mathematics

Volume 32 (2024): Issue 1 (August 2024)

By: Adam Grabowski and Franciszek Turowski

Open Access

|Dec 2024

Abstract

The main aim of this article is to construct two non-trivial examples of weakly associative lattices (also known as trellises). These are generalizations of lattices, not assuming associativity of the lattice operations. We show some connections between trellises and tolerance relations according to the paper of Chajda and Zelinka.

References

Garrett Birkho . Lattice Theory. Providence, Rhode Island, New York, 1967.
Search in Google Scholar Back to article
Ivan Chajda and Bohdan Zelinka. Weakly associative lattices and tolerance relations. Czechoslovak Mathematical Journal, 26(2):259–269, 1976.
Search in Google Scholar Back to article
B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.
Search in Google Scholar Back to article
Ervin Fried and George Grätzer. Some examples of weakly associative lattices. Colloquium Mathematicum, 27:215–221, 1973. doi:10.4064/cm-27-2-215-221.
Open DOI Search in Google Scholar Back to article
Adam Grabowski. On fuzzy negations and laws of contraposition. Lattice of fuzzy negations. Formalized Mathematics, 31(1):151–159, 2023. doi:10.2478/forma-2023-0014.
Open DOI Search in Google Scholar Back to article
Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211–221, 2015. doi:10.1007/s10817-015-9333-5.
Open DOI Search in Google Scholar Back to article
Adam Grabowski. Lattice theory for rough sets – a case study with Mizar. Fundamenta Informaticae, 147(2–3):223–240, 2016. doi:10.3233/FI-2016-1406.
Open DOI Search in Google Scholar Back to article
Adam Grabowski and Takashi Mitsuishi. Extending Formal Fuzzy Sets with Triangular Norms and Conorms, volume 642: Advances in Intelligent Systems and Computing, pages 176–187. Springer International Publishing, Cham, 2018. doi:10.1007/978-3-319-66824-6_16.
Open DOI Search in Google Scholar Back to article
Adam Grabowski and Takashi Mitsuishi. Formalizing lattice-theoretical aspects of rough and fuzzy sets. In D. Ciucci, G. Wang, S. Mitra, and W.Z. Wu, editors, Rough Sets and Knowledge Technology – 10th International Conference held as part of the International Joint Conference on Rough Sets (IJCRS), Tianjin, PR China, November 20–23, 2015, Proceedings, volume 9436 of Lecture Notes in Artificial Intelligence, pages 347–356. Springer, 2015. doi:10.1007/978-3-319-25754-9 31.
Open DOI Search in Google Scholar Back to article
Adam Grabowski and Markus Moschner. Managing heterogeneous theories within a mathematical knowledge repository. In Andrea Asperti, Grzegorz Bancerek, and Andrzej Trybulec, editors, Mathematical Knowledge Management Proceedings, volume 3119 of Lecture Notes in Computer Science, pages 116–129. Springer, 2004. doi:10.1007/978-3-540-27818-4_9. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19–21, 2004.
Open DOI Search in Google Scholar Back to article
Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.
Open DOI Search in Google Scholar Back to article
Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. Equality in computer proof-assistants. In Ganzha, Maria and Maciaszek, Leszek and Paprzycki, Marcin, editor, Proceedings of the 2015 Federated Conference on Computer Science and Information Systems, volume 5 of ACSIS-Annals of Computer Science and Information Systems, pages 45–54. IEEE, 2015. doi:10.15439/2015F229.
Open DOI Search in Google Scholar Back to article
George Grätzer. General Lattice Theory. Academic Press, New York, 1978.
Search in Google Scholar Back to article
George Grätzer. Lattice Theory: Foundation. Birkhäuser, 2011.
Search in Google Scholar Back to article
Yu Kong and Bin Zhao. Uninorms on bounded trellises. Fuzzy Sets and Systems, 481: 108898, 2024. doi:10.1016/j.fss.2024.108898.
Open DOI Search in Google Scholar Back to article
William McCune and Ranganathan Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves. Springer-Verlag, Berlin, 1996.
Search in Google Scholar Back to article
Ranganathan Padmanabhan and Sergiu Rudeanu. Axioms for Lattices and Boolean Algebras. World Scientific Publishers, 2008.
Search in Google Scholar Back to article
Damian Sawicki and Adam Grabowski. On weakly associative lattices and near lattices. Formalized Mathematics, 29(2):77–85, 2021. doi:10.2478/forma-2021-0008.
Open DOI Search in Google Scholar Back to article
Helen Skala. Trellis Theory. Providence, R.I.: American Mathematical Society, 1972.
Search in Google Scholar Back to article
Helen L. Skala. Trellis theory. Algebra Universalis, 1:218–233, 1971. doi:10.1007/BF02944982.
Open DOI Search in Google Scholar Back to article
Lemnaouar Zedam and Bernard De Baets. Triangular norms on bounded trellises. Fuzzy Sets and Systems, 462:108468, 2023. doi:10.1016/j.fss.2023.01.003.
Open DOI Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/forma-2024-0022 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 271 - 279

Accepted on: Dec 27, 2024

Published on: Dec 31, 2024

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

trellis,

tolerance,

weakly associative lattice

Related subjects:

Mathematics,

Logic and set theory,

Computer sciences,

Artificial intelligence

© 2024 Adam Grabowski, Franciszek Turowski, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Volume 32 (2024): Issue 1 (August 2024)