Construction of some algebras of logics by using intuitionistic fuzzy filters on hoops

: In this paper, we define the notions of intuitionistic fuzzy filters and intuitionistic fuzzy implicative (positive implicative, fantastic) filters on hoops. Then we show that all intuitionistic fuzzy filters make a bounded distributive lattice. Also, by using intuitionistic fuzzy filters we introduce a relation on hoops and show that it is a congruence relation, then we prove that the algebraic structure made by it is a hoop. Finally, we investigate the conditions that quotient structure will be di ff erent algebras of logics such as Brouwerian semilattice, Heyting algebra and Wajesberg hoop.


Introduction
Hoops are algebraic structure which are introduced by B. Bosbach in [5,6] are naturally ordered commutative residuated integral monoids. In the last years, some mathematician studied hoop theory in different fields [1-3, 5-10, 17, 19]. Most of these results have a very deep relation with fuzzy logic. Particularly, by using of some theorems and notions of finite basic hoops, in [1] the authors could find a short proof for completeness theorem for propositional basic logic, which is introduced by Hájek in [11]. BL-algebras, the familiar cases of hoops, are the algebraic structures corresponding to basic logic. In algebra of logics, there are some sub-algebras that are very important and have a fundamental role in these algebraic structures. They are very similar to normal subgroups in group theory and ideals in ring theory which we called them filters and by using these notion we can introduce a congruence relation on algebraic structures and study the quotient structure that is made by them. Kondo, in [14], introduced different kinds of filters such as implicative, positive implicative and fantastic filters of hoops and investigated some properties of them. Borzooei and Aaly Kologani in [7], investigated these filters deeply and they introduced some equivalent characterizations of these filters on hoops. Also, they studied the relation among these filters and they showed some equivalent characterizations of these filters.
Zadeh in [21] introduced the notion of fuzzy sets and different kinds of operations on them. After that mathematicians studied on them and applied it to diverse fields. Actually, fuzzy mathematics have reached by studying of fuzzy subsets and their application to mathematical contexts. Nowadays, fuzzy algebra is an important branch of mathematics and mathematicians studied it in different fileds. For example, Rosenfeld in [18] studied fuzzy sub-groups in 1971. After using the concept of fuzzy sets to group theory and defined the notion of fuzzy subgroups, in [21] by Rosenfeld, the different fuzzy algebraic concepts has been growing very fast [12,13,16] and applied in other algebraic structures such as lattices, semigroups, rings, ideals, modules and vector spaces. Moreover, the concepts related to fuzzy sets have been used in various fields, including its use in fuzzy graphs and its application in decision theory. Borzooei and Aaly Kologani in [8], studied the notions of fuzzy filter of hoops and the relation among them and characterized some properties of them. Also, they defined a congruence relation on hoops by a fuzzy filter and proved that the quotient structure of this relation is a hoop.
Atanassov for the first time introduced the term, an intuitionistic fuzzy set [4] that is an extended form of a fuzzy set. These are the sets containing elements having degrees of membership and nonmembership. Intuitionistic fuzzy sets are more adaptable and real intending to the uncertainty and vagueness than the conventional fuzzy sets. The foremost critical property of intuitionistic fuzzy sets not shared by the fuzzy sets is that modular operators can be characterized over intuitionistic fuzzy sets. The intuitionistic fuzzy sets have basically higher depicting conceivable outcomes than fuzzy sets. Also, there are a lot of applications of intuitionistic fuzzy sets in decision making, pattern recognition, medical diagnosis, neural models, image processing, market prediction, color region extraction, and others. In the last years, some mathematician studied intuitionistic fuzzy sets in different fields [15,20]. In decision-making problems, the use of fuzzy approaches is ubiquitous. The purpose of these intuitionistic fuzzy sets is, to provide a new approach with useful mathematical tools to address the fundamental problem of decision-making. The generality of the fuzzy set is given special importance, illustrating how many interesting decision-making problems can be formulated as a problem of fuzzy sets. These applied contexts provide solid evidence of the wide applications of fuzzy sets approach to model and research decision-making problems. Now, in the following we define the notions of intuitionistic fuzzy filters and intuitionistic fuzzy implicative (positive implicative, fantastic) filters on hoops. Then we show that all intuitionistic fuzzy filters make a bounded distributive lattice. Also, by using intuitionistic fuzzy filters we introduce a relation on hoops and show that it is a congruence relation, then we prove that the algebraic structure made by it is a hoop. Finally, we investigate the conditions that quotient structure will be different algebras of logics such as Brouwerian semilattice, Heyting algebra and Wajesberg hoop.

Preliminaries
In this section, we refer to the basic concepts and properties required in the field of hoop and fuzzy sets that we will use in the following sections.
A hoop is an algebraic structure (H, * , ↠, 1) such that, for any d, s, q ∈ H: On hoop H, we define the relation ⪯ by d ⪯ s iff d ↠ s = 1. Obviously, (H, ⪯) is a poset. A hoop H is said to be bounded if H has a least element such as 0, it means that for all d ∈ H, we have 0 ⪯ d.
If H is bounded, then we can introduce a unary operation " ′ "on H such that d ′ = d ↠ 0, for any d ∈ H. The bounded hoop H is said to have the double negation property, or (DNP), for short if (d ′ ) ′ = d, for any d ∈ H (see [1]).
The next proposition provides some properties of hoop.
Proposition 2.1. [5,6] Suppose (H, * , ↠, 1) is a hoop and d, s, q ∈ H. Then: Proposition 2.2. [5,6] Suppose H is a bounded hoop and d, s ∈ H. Then: Then for any d, s, q ∈ H, the next conditions are equivalent:  [10,14] The family of all intuitionistic fuzzy sets on H will be denoted by IF S(H).
and C = (ς C , ϱ C ) be two IF-sets on H. Then, for any d ∈ H, we define a relation between them as follows: A fuzzy set ς on hoop H is called a fuzzy filter of H if for all d, s ∈ H, ς(d) ⪯ ς(1) and Notation. In the following, we will consider in this article H as a hoop and ς and ϱ as fuzzy sets. Moreover, the set of all fuzzy filters of H and anti-fuzzy filter of H are denoted by FF(H) and AFF(H), respectively.

Construction of hoops by anti-fuzzy filters
In this section, the concept of anti-fuzzy filter on hoop H is defined and some related results are investigated.
A complement of ς is the fuzzy set ς c which is defined by, ς c (d) = 1 − ς(d), for any d ∈ H.
Proof. Since ϱ ∈ AFF(H), for all d ∈ H, we get Moreover, since ϱ is an anti-fuzzy filter, by Propositions 2.1(x), 3.5(iii) and Remark 3.3(2), we have So, d ≈ ϱ q and this means that ≈ ϱ is transitive. Hence, ≈ ϱ is an equivalence relation. Now, we prove that ≈ ϱ is a congruence relation. .
For any e ∈ H, [e] ϱ denotes the equivalence class of e with respect to ≈ ϱ . Clearly ϱ | e ∈ H} and operations ⊗ and → on H ≈ ϱ defined as follows: Then Since ≈ ϱ is the congruence relation on H, we get that all above operations are well-defined. Thus, by routine calculation, we can see that H ≈ ϱ is a hoop. Now, we define a binary relation on Easily we can see ( H ≈ ϱ , ⪯) is a partial order monoid. □

Construction of hoops and distributive lattices by intuitionistic fuzzy filters
In the following, we define the concept of intuitionistic fuzzy filter on hoop H and investigate some related results.
The family of all intuitionistic fuzzy filters of H will be denoted by IF F (H).
is an intuitionistic fuzzy filter on H.
Proof. By Definitions 3.1, 4.1 and Proposition 2.5, the proof is clear. □ In the next example, we show that the condition In the next proposition we prove that by a fuzzy filter (anti-fuzzy filter) on A we can make an IF-filter.
be an IF-set on H and for any r ∈ [0, 1], Then B is an IF-filter of H iff for any r ∈ [0, 1], B r ∅ is a filter of H.
Proof. By Theorems 2.6, 3.4 and Proposition 4.3, the proof is clear. □ Now, we mainly investigate the lattice of all IF-filters by introducing the notion of tip-extended pair of IF-sets.
be a set of IF-filters of H and fuzzy sets i∈I ς B i and i∈I ϱ B i on H are defined as follows: Let L be an IF-set on H. The intersection of all IF-filters containing L is called the generated IF-filter by L, denoted as ⟨L⟩.
Then ≈ B is a congruence relation on H.
Proof. The proof is similar to the proof of Theorems 2.7 and 3.6. □

Construction of Brouwerian semilattice by intuitionistic fuzzy (positive) implicative filters
Here, we define the notions of intuitionistic fuzzy (positive) implicative filters on hoops and some related results are investigated. We find some equivalence characterizations of them.
Definition 5.1. An IF-set B = (ς B , ϱ B ) on H is called an intuitionistic fuzzy implicative filter or an IF-implicative filter of H if for all d, s, q ∈ H, it satisfies the next conditions: Proof. Let B = (ς B , ϱ B ) be an IF-implicative filter. By (IFIF 1 ), obviously (IFF1) holds. Since d * s ⪯ d * s, by Proposition 2.1(ii), s ⪯ d ↠ (d * s). Then by (IFIF 1 ), ϱ B (d ↠ (d * s)) ⪯ ϱ B (s). Also, since ϱ B is an IF-implicative filter, it is enough to choose q = 1, then Similarly, we can see In the next example we can see that the converse of the previous theorem does not hold.
Proof. In this proof, we just prove the items of ϱ B . Also, since B is an IF-filter of H, by Definition 4.1, ς B ∈ FF(H) and ϱ B ∈ AFF(H). Then as we notice, ϱ B (1) ⪯ ϱ B (d) and ς B (1) ⪰ ς B (d), for any d ∈ H.
(i) ⇒ (ii) It is enough to let d = 1 in (IFIF 2 ). Then Hence, B is an IF-implicative filter of H.
Also, by Proposition 2.1(ii), d ⪯ s ↠ (d * s). Then by ( Therefore, B is an IF-filter of H. □ Next example shows that the converse of the previous theorem does not hold. Example 5.11. According to Example 5.2, introduce ς B by ς B (1) = r 3 , ς B (k) = ς B (e) = r 2 and ς B (0) = r 1 such that 0 ⪯ r 1 < r 2 < r 3 ⪯ 1. Routine calculation shows ς B ∈ FF(H) and is not an IF-positive implicative filter. Because Theorem 5.12. Let B = (ς B , ϱ B ) be an IF-filter of H. Then for any d, s, q ∈ H the next conditions are equivalent: Proof. In this proof, we just prove the items of ϱ B . Also, since B is an IF-filter of H, by Definition 4.1, ς B ∈ FF(H) and ϱ B ∈ AFF(H). Then as we notice, ϱ B (1) ⪯ ϱ B (d) and ς B (1) ⪰ ς B (d), for any d ∈ H.
As we prove, By the similar way, ς B (d ↠ d 2 ) = ς B (1). Hence, by Theorem 5.12(v), B is an IF-positive implicative filter. □ The next example shows that the converse of the pervious theorem does not hold, in general.
Since H has (DNP), by (HP3), we have, By the similar way, Moreover, since B is an IF-positive implicative filter of H, by Theorem 5.12(iii), Proposition 2.2(ii) and (DNP), we get By the similar way, . Therefore, by Theorem 5.6(ii), B is an IF-implicative filter.
Thus, we have, Since B is an IF-implicative filter, by Theorem 5.4, B is an IF-filter. Thus, by Theorem 5.6(ii), Since B is an IF-positive implicative filter, by Theorem 5.12(ii), we have Also, by assumption, On the other hand, by Proposition 2.1(iv), s ⪯ d ↠ s and by Proposition 2.1(xiii), (d ↠ s) ↠ d ⪯ s ↠ d. Since B is an IF-positive implicative filter, by Theorem 5.10, B is an IF-filter, thus, Since B is an IF-positive implicative filter, by Theorem 5.6(iii), we have

Construction of Heyting semilattice and Wajesberg hoops by intuitionistic fuzzy fantastic filters
In the following, the concept of intuitionistic fuzzy fantastic filter on hoops is defined and some related results are investigated.
Example 6.2. According to Example 5.5, routine calculation shows that B is an IF-fantastic filter.
By the similar way, we can see that ς B (d) ∧ ς B (d ↠ s) ⪯ ς B (s). Therefore, B is an IF-filter of H. □ In the next example we show that the converse of the previous theorem does not hold, in general.
Example 6.5. According to Example 5.5, B is an IF-filter but it is not an IF-fantastic filter. Because Theorem 6.6. Let B = (ς B , ϱ B ) be an IF-filter of H. Then the next statements are equivalent, for all d, s ∈ H, (i) B is an IF-fantastic filter of H, Proof. In this proof, we just prove the items of ϱ B . Also, since B is an IF-filter of H, by Definition 4.1, ς B ∈ FF(H) and ϱ B ∈ AFF(H). Then as we notice, ϱ B (1) ⪯ ϱ B (d) and ς B (1) ⪰ ς B (d), for any d ∈ H.
(ii) ⇒ (i) Let d, s, q ∈ H. Since B is an IF-filter, we get ϱ B (s ↠ d) ⪯ ϱ B (q) ∨ ϱ B (q ↠ (s ↠ d)). Thus, by (ii), Since B is an IF-filter, Now, since B is an IF-fantastic filter, we have Then by (HP3), Moreover, since B is an IF-implicative filter, Hence, by Theorem 6.6(ii), B is an IF-fantastic filter. □ The next example shows that the converse of the pervious theorem may not be true, in general.
Since B is an IF-positive implicative filter, by Theorem 5.10, B is an IF-filter of H, we have Also, since B is an IF-positive implicative filter, by Theorem 5.12(ii), we get On the other side, by Proposition 2.1(iv) and (xiii), . Moreover, since B is an IF-fantastic filter, by Theorem 6.6(ii), From B is an IF-filter, then Thus, by Theorem 5.6(ii), B is an IF-implicative filter. □

Conclusions
In decision problems, the use of fuzzy approaches is ubiquitous. Given the importance of fuzzy concepts in solving decision problems, we decided to use these concepts, intuitionistic fuzzy sets, in a specific logical algebra to provide a new approach with useful mathematical tools to address the fundamental decision problem. In this paper, the concept of anti-fuzzy filter of hoops is defined and the concepts of intuitionistic fuzzy filters, intuitionistic fuzzy (positive) implicative and intuitionistic fuzzy fantastic filters of hoops are introduced and the properties and equivalent characterizations of them are discussed. Moreover, it was proved that all intuitionistic fuzzy filters make a bounded distributive lattice. Also, the relations between different kinds of intuitionistic fuzzy filters are investigated and studied that under which conditions they are equivalent. Also, a congruence relation on hoops is defined by an intuitionistic fuzzy filter and proved the new structure is a hoop. Finally, the conditions that quotient structure will be Brouwerian semilattice, Heyting algebra and Wajesberg hoop are investigated.