The degree of reflexivity of fuzzy relations in terms of left-continuous t-norms

In this paper, we explore the degree of reflexivity of fuzzy relations. We mainly focus on exploring two new types of the degree of reflexivity of fuzzy relations (we briefly call them as the type-II degree of reflexivity and the type-III degree of reflexivity) that are different from the existing degree of reflexivity of fuzzy relations in the literature (we briefly call it as the type-I degree of reflexivity). The notion of the type-II degree of reflexivity is introduced through the use of the type-I degree of reflexivity and R-implications derived from left-continuous t-norms, while the notion of the type-III degree of reflexivity is introduced through the use of fuzzy equalities based on R-implications derived from left-continuous t-norms. We also illustrate that these three types of the degree of reflexivity are not equal to each other in general, and further characterize when they are equal to each other. Moreover, it is shown that the type-I degree of reflexivity and the type-III degree of reflexivity preserve fuzzy equalities, while the type-II degree of reflexivity does not preserve fuzzy equalities in general. We also characterize when the type-II degree of reflexivity preserves fuzzy equalities. Finally, we show that for continuous t-norms or the nilpotent minimum t-norm, and for a given fuzzy relation R on a set X, there exists a sub-epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-reflexive (epsilon is an element of[0,1])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon \in [0,1])$$\end{document} fuzzy relation S on X, which is close enough to R with respect to fuzzy equalities, such that the degree of the fuzzy equality between S and R is the type-III degree of reflexivity of R.