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Quasi-pseudo-BL algebras and weak filters

Soft Computing volume 27, pages 2185–2204 (2023)Cite this article

Abstract

In this paper, we introduce the notion of weak filters in quasi-pseudo-BL algebras. First, we discuss the properties of weak filters of a quasi-pseudo-BL algebra and characterize the weak filters generated by some non-empty subsets. Second, we investigate the relationship between weak filters of a quasi-pseudo-BL algebra and filters of its pseudo-BL subalgebra. Moreover, we study some kinds of weak filters such as normal, prime, maximal and weak prime weak filters. Third, the quotient quasi-pseudo-BL algebras with respect to normal weak filters are defined and the relation between (weak) filters of a quasi-pseudo-BL algebra and (weak) filters of its associated quotient algebra is discussed. Finally, we present the topological properties of the set of all prime weak filters of a quasi-pseudo-BL algebra and show that the prime spectrum is a complete \(T_{0}\) topological space.

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References

  • Bou F, Paoli F, Ledda A, Spinks M, Giuntini R (2010) The logic of quasi-MV algebras. J Log Comput 20:619–643

    Article  MathSciNet  MATH  Google Scholar 

  • Chajda I (1992) Lattices in quasiordered sets, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 31:6–12

  • Chajda I (1994) An algebra of quasiordered logic. Math Biochem 119:129–135

    MathSciNet  MATH  Google Scholar 

  • Chang CC (1959) A new proof of the completeness of Lukasiewicz axioms. Trans Am Math Soc 93:74–80

    MATH  Google Scholar 

  • Chen WJ, Dudek WA (2016) Quantum computational algebra with a non-commutative generalization. Mathmatics Slovaca 66:19–34

    Article  MathSciNet  MATH  Google Scholar 

  • Chen WJ, Dudek WA (2018) Ideals and congruences in quasi-pseudo-MV algebras. Soft Comput 22:3879–3889

    Article  MATH  Google Scholar 

  • Chen WJ, Chen ZY, Wang HK (2022) Quasi-pseudo-hoops: an extension to Pseudo-hoops. J Multiple Valued Logic Soft Comput 38:333–353

    MATH  Google Scholar 

  • Chen WJ, Wang HK (2020) Filters and ideals in the generalization of pseudo-BL algebras. Soft Comput 24:795–812

    Article  MATH  Google Scholar 

  • Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo BL-algebras: part I. J Multiple Valued Logic Soft Comput 8:673–714

    MATH  Google Scholar 

  • Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo BL-algebras: part II. J Multiple Valued Logic Soft Comput 8:715–750

    MATH  Google Scholar 

  • Georgescu G, Leustean L (2002) Some classes of pseudo-BL algebras. J Aust Math Soc 73:127–153

    Article  MathSciNet  MATH  Google Scholar 

  • Hajek P (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  • Iorgulescu A (2018) Implicative-groups vs. groups and generalizations. Matrix Rom, Bucuresti

    MATH  Google Scholar 

  • Kowalski T, Paoli F (2011) Joins and subdirect products of varieties. Algebra Universalis 65:371–391

    Article  MathSciNet  MATH  Google Scholar 

  • Kowalski T, Paoli F, Spinks M (2011) Quasi-subtractive varieties. J Symb Log 76:1261–1286

    Article  MathSciNet  MATH  Google Scholar 

  • Ledda A, Konig M, Paoli F, Giuntini R (2006) MV algebras and quantum computation. Stud Logica 82:245–270

    Article  MathSciNet  MATH  Google Scholar 

  • Turunen E (1999) Mathematics behind fuzzy logic. Physica-Verlag, Heidelberg

    MATH  Google Scholar 

Download references

Funding

This work was supported by Shandong Provincial Natural Science Foundation (Grant Number ZR2020MA041), China Postdoctoral Science Foundation (Grant Number 2017M622177) and Shandong Province Postdoctoral Innovation Projects of Special Funds (Grant Number 201702005).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Jinan, No. 336, West Road of Nan Xinzhuang, Jinan, 250022, Shandong, People’s Republic of China

    Wenjuan Chen & Jinfan Xu

Authors
  1. Wenjuan Chen
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  2. Jinfan Xu
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Contributions

Wenjuan Chen contributed to the study conception and design. The first draft of the manuscript was written by WC and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Wenjuan Chen.

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Chen, W., Xu, J. Quasi-pseudo-BL algebras and weak filters. Soft Comput 27, 2185–2204 (2023). https://doi.org/10.1007/s00500-022-07744-y

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  • DOI: https://doi.org/10.1007/s00500-022-07744-y

Keywords

  • Filters
  • Weak filters
  • Quasi-pseudo-BL algebras
  • Pseudo-BL algebras