Operators on complemented lattices.
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| Title: | Operators on complemented lattices. |
|---|---|
| Authors: | Chajda, Ivan1 (AUTHOR) ivan.chajda@upol.cz, Länger, Helmut1,2 (AUTHOR) helmut.laenger@tuwien.ac.at |
| Source: | Soft Computing - A Fusion of Foundations, Methodologies & Applications. Apr2025, Vol. 29 Issue 7, p3115-3123. 9p. |
| Subjects: | Propositional calculus, Quantum mechanics, Definitions |
| Abstract: | The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several complements. The mapping + assigning to each element a the set a + of all its complements is investigated as an operator on the given lattice. We can extend the definition of a + in a natural way from elements to arbitrary subsets. In particular we study the set a + for complemented modular lattices, and we characterize when the set a + + is a singleton. By means of the operator + we introduce two other operators → and ⊙ which can be considered as implication and conjunction in a certain propositional calculus, respectively. These two logical connectives are "unsharp" which means that they assign to each pair of elements a non-empty subset. However, also these two derived operators share a lot of properties with the corresponding logical connectives in intuitionistic logic or in the logic of quantum mechanics. In particular, they form an adjoint pair. Finally, we define so-called deductive systems and we show their relationship to the mentioned operators as well as to lattice filters. [ABSTRACT FROM AUTHOR] |
| Copyright of Soft Computing - A Fusion of Foundations, Methodologies & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 185783279 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Operators on complemented lattices. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Chajda%2C+Ivan%22">Chajda, Ivan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ivan.chajda@upol.cz</i><br /><searchLink fieldCode="AR" term="%22Länger%2C+Helmut%22">Länger, Helmut</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> helmut.laenger@tuwien.ac.at</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Soft+Computing+-+A+Fusion+of+Foundations%2C+Methodologies+%26+Applications%22">Soft Computing - A Fusion of Foundations, Methodologies & Applications</searchLink>. Apr2025, Vol. 29 Issue 7, p3115-3123. 9p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Propositional+calculus%22">Propositional calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+mechanics%22">Quantum mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Definitions%22">Definitions</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several complements. The mapping + assigning to each element a the set a + of all its complements is investigated as an operator on the given lattice. We can extend the definition of a + in a natural way from elements to arbitrary subsets. In particular we study the set a + for complemented modular lattices, and we characterize when the set a + + is a singleton. By means of the operator + we introduce two other operators → and ⊙ which can be considered as implication and conjunction in a certain propositional calculus, respectively. These two logical connectives are "unsharp" which means that they assign to each pair of elements a non-empty subset. However, also these two derived operators share a lot of properties with the corresponding logical connectives in intuitionistic logic or in the logic of quantum mechanics. In particular, they form an adjoint pair. Finally, we define so-called deductive systems and we show their relationship to the mentioned operators as well as to lattice filters. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Soft Computing - A Fusion of Foundations, Methodologies & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00500-025-10626-8 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 9 StartPage: 3115 Subjects: – SubjectFull: Propositional calculus Type: general – SubjectFull: Quantum mechanics Type: general – SubjectFull: Definitions Type: general Titles: – TitleFull: Operators on complemented lattices. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Chajda, Ivan – PersonEntity: Name: NameFull: Länger, Helmut IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 14327643 Numbering: – Type: volume Value: 29 – Type: issue Value: 7 Titles: – TitleFull: Soft Computing - A Fusion of Foundations, Methodologies & Applications Type: main |
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