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We use a neutrosophic set, instead of an intuitionistic fuzzy because the neutrosophic set is more general, and it allows for independent and partial independent components μ (χ) , γ (χ) , ζ (χ), while in an intuitionistic fuzzy set, all components are totally dependent. In this article, we present and demonstrate the concept of neutrosophic invariant subgroups. We delve into the exploration of this notion to establish and study the neutrosophic quotient group. Further, we give the concept of a neutrosophic normal subgroup as a novel concept. [ABSTRACT FROM AUTHOR]

Considering L and M being different commutative quantales, this paper mainly proposes a many-level version of L-fuzzy generalized neighborhood systems and then a pair of many level upper and lower rough approximation operators based on it are defined and discussed. It is proved that these approximation operators include L-fuzzy generalized neighborhood-based approximation operators and M-level L-fuzzy relation-based approximation operators as their special circumstances. Moreover, we define the measure of many level rough approximations that in a certain sense characterizes the quality of the obtained approximation. Furthermore, when the M-level L-fuzzy generalized neighborhood system is non-increasing, serial, reflexive, transitive, and unary, then the corresponding M-level L-rough approximation operators are discussed and characterized, respectively. Moreover, the relationship between M-level L-rough approximation operators and (L , M) -quasi-fuzzy topologies is presented. [ABSTRACT FROM AUTHOR]

In this paper we consider the category M-FQMod of M-fuzzy (left) Q-modules over a given quantale Q. We introduce and investigate the notion of M-fuzzy Q-submodule of a left Q-module. Also, the notions of M-fuzzy module nuclei and conuclei on a left Q-module are defined, and the elementary properties of both are studied. The relationships between such notions and M-fuzzy Q-submodule are discussed. Finally, the relationship between the category of M-fuzzy Q-modules and the category of (L, M)-quasi-fuzzy topological spaces is discussed. [ABSTRACT FROM AUTHOR]