Exploring fuzzy filters and their relationships on Sheffer stroke basic algebras

This article introduces the concepts of fuzzy filters, the generated fuzzy filters, fuzzy prime filters, fuzzy compatible filters, and the cosets of a fuzzy filter on Sheffer stroke basic algebras. It puts forward various characteristics and relationships among these concepts. The study reveals that the minimum of fuzzy filters is also a fuzzy filter, and it constructs the smallest fuzzy filter covering a fuzzy set. The generation of fuzzy filters by fuzzy sets is investigated, and it is demonstrated that the minimum of fuzzy filters generated by two different fuzzy sets equals the fuzzy filter generated by the minima of these sets. Furthermore, the extension theorem of fuzzy prime filters is established, along with the necessary condition for the minimum of two fuzzy prime filters to be a fuzzy prime filter. Finally, the structure (B/μB,|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(B/\mu _B,|)$$\end{document} is identified as a Sheffer stroke basic algebra, and the Ker¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{Ker}$$\end{document} mapping is introduced to establish the isomorphism between B/μB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B/\mu _B$$\end{document} and B/Ker¯.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B/\overline{Ker}.$$\end{document}