Abstract
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.
Highlights
Hoop is introduced by blueBosbach in [1], and it is naturally ordered commutative residuated integral monoids and he investigated some properties of it in [2]
The idea of quasi-coincidence of a fuzzy point with a fuzzy set is mentioned in [12], and it played a vital role to generate some different types of fuzzy subalgebras in of BCK/BCI-algebras, called on (α, β)-fuzzy subalgerbas of BCK/BCI-algebras which is introduced by Jun [13]
In [19], Aaly introduced the notions of (∈, ∈)-fuzzy filters and (∈, ∈ ∨q)-fuzzy filters on hoops and disscussed some properties of them. They used these notions and defined a congruence relation on hoops and proved that the quotient structure that is made by this relation is a hoop
Summary
Hoop is introduced by blueBosbach in [1], and it is naturally ordered commutative residuated integral monoids and he investigated some properties of it in [2]. BlueBorzooei and Aaly Kologani in [5] defined (implicative, positive implicative, fantastic) filters in a hoop and discussed their relations and properties Using filter, they considered a congruence relation on a hoop, and induced the quotient structure which is a hoop. In [19], Aaly introduced the notions of (∈, ∈)-fuzzy filters and (∈, ∈ ∨q)-fuzzy filters on hoops and disscussed some properties of them They used these notions and defined a congruence relation on hoops and proved that the quotient structure that is made by this relation is a hoop. It is natural to consider similar style of generalizations of the existing fuzzy subsystems of other algebraic structures For this reason, we decided to define and investigated these notions on hoop algebras, which we studied [20,21,22,23] for sources of inspiration and ideas for this paper. We defined some equivalent definitions of them and used the congruence relation on hoop defined in [19] by an (∈, ∈)-fuzzy filter of hoop, and proved that the quotient structure that is made by this relation is a Brouwerian semilattice
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