On α-state filters in state residuated lattices.
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| Title: | On α-state filters in state residuated lattices. |
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| 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}$| -space or a Hausdorff space. In addition, some topological characterizations are summarized for a state residuated lattice to be quasicomplemented. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | The purpose of this paper is to investigate |$\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] |
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| ISSN: | 0955792X |
| DOI: | 10.1093/logcom/exae078 |
