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The aim of this paper is defining the concepts of Noetherian and Artinian hoops by using the filter of hoop in the partial order set of all the filters of hoops and inclusion relation and find some equivalent definitions for this notion. We translate some important results from theory of rings to the case of hoop and their characterizations are established. The relation between short exact sequence on Noetherian and Artinian hoop studied and by using short exact sequence we prove that the Cartesian product of two hoops is Noetherian (Artinian) if and only if each one is a Noetherian (Artinian). By using the notion of filter in hoops, we define the notion of composition series and prove any V-hoop is Noetherian and Artinian if and only if it has composition series. Finally, Chinese Remainder theorem in hoop and the relation between maximal filter and Noetherian (Artinian) hoop are investigated. [ABSTRACT FROM AUTHOR]

When it comes to making decisions in vague problems, Rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations-lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop. [ABSTRACT FROM AUTHOR]

The falling shadow theory is applied to subhoops and filters in hoops. The notions of falling fuzzy subhoops and falling fuzzy filters in hoops are introduced, and several properties are investigated. Relationship between falling fuzzy subhoops and falling fuzzy filters are discussed, and conditions for a falling fuzzy subhoop to be a falling fuzzy filter are provided. Also conditions for a falling shadow of a random set to be a falling fuzzy filter are displayed. [ABSTRACT FROM AUTHOR]

In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a V-hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship between ideals and filters by exploiting the set of complements. [ABSTRACT FROM AUTHOR]

Hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach in [6, 7], that BL-algebras are particular cases of hoops. Now, in this paper, we introduce the concept of (semi)topological hoop and we get some related results. Then we derive here conditions that imply a hoop to be a semitopological or a topological hoop and we study some properties of them. Specially, we show that in a hoop A, if (A; → T) is a semitopological hoop and f1g is an open set or A is bounded and satisfies the double negation property, then (A; T) is a topological hoop. Finally, we construct a discrete topology on quotient hoops, under suitable conditions. [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]