Fuzzy Filters of Hoops Based on Fuzzy Points

Abstract

In this paper, we define the concepts of ( ∈ , ∈ ) and ( ∈ , ∈ ∨ q ) -fuzzy filters of hoops, discuss some properties, and find some equivalent definitions of them. We define a congruence relation on hoops by an ( ∈ , ∈ ) -fuzzy filter and show that the quotient structure of this relation is a hoop.

Fuzzy sub-hoops based on fuzzy points

“…Also, we found relations between an (∈, ∈)-fuzzy sub-hoop, an (∈, ∈ ∨ q)-fuzzy sub-hoop and a (q, ∈ ∨ q)-fuzzy sub-hoop and considered conditions for a fuzzy set to be a (q, ∈ ∨ q)-fuzzy sub-hoop of H, and provided a condition for an (∈, ∈ ∨ q)-fuzzy subhoop to be a (q, ∈ ∨ q)-fuzzy sub-hoop. By [1,6,7,14] we defined the concept of (∈, ∈)-fuzzy filters (fuzzy implicative filters, fuzzy positive implicative filters, fuzzy fantastic filters) of hoop and (∈, ∈ ∨ q)-fuzzy filters (fuzzy implicative filters, fuzzy positive implicative filters, fuzzy fantastic filters) of hoop and have investigated some equivalent definitions and properties of them.…”

Fuzzy Positive Implicative Filters of Hoops Based on Fuzzy Points

“…Definition 2. [19] Let (α, β) be any one of (∈, ∈) and (∈, ∈ ∨ q). A fuzzy set λ in H is called an (α, β)-fuzzy filter of H if the following conditions hold.…”

“…Note. According to Aaly Kologani et al [19], every (∈, ∈)-fuzzy sub-hoop is an (∈, ∈ ∨ q)-fuzzy sub-hoop of H. Easily we can consequence that every (∈, ∈)-fuzzy positive implicative filter of H is an (∈, ∈ ∨ q)-fuzzy positive implicative filter of H. But by Example ([25] Example 3.9), there exists (∈, ∈ ∨ q)-fuzzy positive implicative filter of H that is not an (∈, ∈ ∨ q)-fuzzy filter. So some of above theorem that proved in this section, hold for (∈, ∈ ∨ q)-fuzzy positive implicative filter of H. But some of them hold with conditions, because of that we prove them again.…”

“…Also, Bakhshi in [17], investigated (α, β)-fuzzy ideals in pseudo MV-algebras and m-polar (α, β)-fuzzy ideals in BCK/BCI-algebras is studied by Al-Masarwah in [18]. Moreover, in [19], Aaly introduced the notions of (∈, ∈)-fuzzy filters and (∈, ∈ ∨q)-fuzzy filters on hoops and disscussed some properties of them. Then 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.…”

“…The purpose of this paper is define the concepts of (∈, ∈)-fuzzy positive implicative filters and (∈, ∈ ∨q)-fuzzy positive implicative filters of hoops, by inspiring the concepts of (∈, ∈)((∈, ∈ ∨q))-fuzzy filters, and some properties of them are investigated. Then 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.…”