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On α-state filters in state residuated lattices.

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Title: On α-state filters in state residuated lattices.
Authors: Wu, Supeng1 (AUTHOR), Liang, Xingliang1 (AUTHOR), Yang, Jiang2 (AUTHOR)
Source: Journal of Logic & Computation. Jun2025, Vol. 35 Issue 4, p1-30. 30p.
Subjects: Residuated lattices, Heyting algebras, Hausdorff spaces, Topology
Abstract: The purpose of this paper is to investigate |$\alpha $| -state filters in a state residuated lattice. First, the notion of |$\alpha $| -state filters in a state residuated lattice is introduced. It is proved that the set of all |$\alpha $| -state filters in a state residuated lattice forms a complete Heyting algebra. Furthermore, the concept of quasicomplemented state residuated lattices is presented. Some equivalent conditions are derived for quasicomplemented state residuated lattices. Finally, the hull-kernel topology on the set of all prime |$\alpha $| -state filters in a state residuated lattice is investigated. It is demonstrated that the set of all prime |$\alpha $| -state filters under the hull-kernel topology is a spectral space and some certain conditions are given for the space to be either a |$T_{1}$| -space or a Hausdorff space. In addition, some topological characterizations are summarized for a state residuated lattice to be quasicomplemented. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Jun2025, Vol. 35 Issue 4, p1-30. 30p.
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  Data: The purpose of this paper is to investigate |$\alpha $| -state filters in a state residuated lattice. First, the notion of |$\alpha $| -state filters in a state residuated lattice is introduced. It is proved that the set of all |$\alpha $| -state filters in a state residuated lattice forms a complete Heyting algebra. Furthermore, the concept of quasicomplemented state residuated lattices is presented. Some equivalent conditions are derived for quasicomplemented state residuated lattices. Finally, the hull-kernel topology on the set of all prime |$\alpha $| -state filters in a state residuated lattice is investigated. It is demonstrated that the set of all prime |$\alpha $| -state filters under the hull-kernel topology is a spectral space and some certain conditions are given for the space to be either a |$T_{1}$| -space or a Hausdorff space. In addition, some topological characterizations are summarized for a state residuated lattice to be quasicomplemented. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA 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:
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    Identifiers:
      – Type: doi
        Value: 10.1093/logcom/exae078
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 30
        StartPage: 1
    Subjects:
      – SubjectFull: Residuated lattices
        Type: general
      – SubjectFull: Heyting algebras
        Type: general
      – SubjectFull: Hausdorff spaces
        Type: general
      – SubjectFull: Topology
        Type: general
    Titles:
      – TitleFull: On α-state filters in state residuated lattices.
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            NameFull: Wu, Supeng
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          Name:
            NameFull: Liang, Xingliang
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            NameFull: Yang, Jiang
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          Dates:
            – D: 01
              M: 06
              Text: Jun2025
              Type: published
              Y: 2025
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              Value: 35
            – Type: issue
              Value: 4
          Titles:
            – TitleFull: Journal of Logic & Computation
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