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In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization of abstract nilpotent groups. For this, we show that a group G is nilpotent if and only if any fuzzy subgroup of G is a g-nilpotent fuzzy subgroup of G. In particular, we construct a nilpotent group via a g-nilpotent fuzzy subgroup. Finally, we characterize the elements of any maximal normal abelian subgroup by using a g-nilpotent fuzzy subgroup. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the notion of an m-polar fuzzy (normal) subalgebra in B-algebras and its related properties. We consider characterizations of an m-polar fuzzy (normal) subalgebra. We define the concept of an m-polar intuitionistic fuzzy (normal) subalgebra in a B-algebra, and we characterize the m-polar intuitionistic fuzzy (normal) subalgebra. Given an m-polar fuzzy set, we construct a simple m-polar fuzzy set and discuss on m-polar intuitionistic fuzzy subalgebras of B-algebras. [ABSTRACT FROM AUTHOR]

The notion of (normal) m-polar (∈ , ∈) -fuzzy p-ideals of BCI-algebras is introduced, and several properties are investigated. Relations between an m-polar (∈ , ∈) -fuzzy ideal and an m-polar (∈ , ∈) -fuzzy p-ideal are displayed, and conditions for an m-polar (∈ , ∈) -fuzzy ideal to be an m-polar (∈ , ∈) -fuzzy p-ideal are provided. Characterization of m-polar (∈ , ∈) -fuzzy p-ideals are considered. Given an m-polar (∈ , ∈) -fuzzy ideal (resp., m-polar (∈ , ∈) -fuzzy p-ideal), a normal m-polar (∈ , ∈) -fuzzy ideal (resp., normal m-polar (∈ , ∈) -fuzzy p-ideal) is established. Using an m-polar (∈ , ∈) -fuzzy ideal, the quotient structure of BCI-algebras is constructed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of (∈ , ∈) -fuzzy positive implicative filters and (∈ , ∈ ∨ q) -fuzzy positive implicative filters in hoops and investigate their properties. We also define some equivalent definitions of them, and then we use the congruence relation on hoop defined in blue [Aaly Kologani, M.; Mohseni Takallo, M.; Kim, H.S. Fuzzy filters of hoops based on fuzzy points. Mathematics.2019, 7, 430; doi:10.3390/math7050430] by using an (∈ , ∈) -fuzzy filter in hoop. We show that the quotient structure of this relation is a Brouwerian semilattice. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using zero divisor elements, we construct a productive graph and prove that it is connected and both complete and a tree under some conditions. [ABSTRACT FROM AUTHOR]