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Weak Inflationary BL-Algebras and Filters of Inflationary (Pseudo) General Residuated Lattices

Author

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  • Xiaohong Zhang

    (Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China
    Shaanxi Joint Laboratory of Artificial Intelligence, Shaanxi University of Science & Technology, Xi’an 710021, China)

  • Rong Liang

    (Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China)

  • Benjamín Bedregal

    (Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal 59084-100, RN, Brazil)

Abstract

After the research on naBL-algebras gained by the non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of non-associative generalization of BL-algebras, which can be obtained by general overlap functions. In this paper, we show that not every inflationary general overlap function can induce an inflationary BL-algebra by a counterexample and thus propose the new concept of weak inflationary BL-algebras. We prove that each inflationary general overlap function corresponds to a weak inflationary BL-algebra; therefore, two mistaken results in the previous paper are revised. In addition, some properties satisfied by weak inflationary BL-algebras are discussed, and the relationships among some non-classical logic algebras are analyzed. Finally, we establish the theory of filters and quotient algebras of inflationary general residuated lattice (IGRL) and inflationary pseudo-general residuated lattice (IPGRL), and characterize the properties of some kinds of IGRLs and IPGRLs by naBL-filters, (weak) inflationary BL-filters, and weak inflationary pseudo-BL-filters.

Suggested Citation

  • Xiaohong Zhang & Rong Liang & Benjamín Bedregal, 2022. "Weak Inflationary BL-Algebras and Filters of Inflationary (Pseudo) General Residuated Lattices," Mathematics, MDPI, vol. 10(18), pages 1-21, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:18:p:3394-:d:918563
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    References listed on IDEAS

    as
    1. Nan Sheng & Xiaohong Zhang, 2022. "Regular Partial Residuated Lattices and Their Filters," Mathematics, MDPI, vol. 10(14), pages 1-16, July.
    2. Rong Liang & Xiaohong Zhang, 2022. "Pseudo General Overlap Functions and Weak Inflationary Pseudo BL-Algebras," Mathematics, MDPI, vol. 10(16), pages 1-15, August.
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