Research Article On Fuzzy Deductive Systems of Hilbert Algebras

. In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the inﬁnite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.


Introduction
e pioneering work of Zadeh [1] on fuzzy subsets of a set has been extensively applied to many scientific fields. is concept was adapted by Rosenfeld [2] to define fuzzy subgroups of a group. Since then, many authors have been studying fuzzy subalgebras of several algebraic structures such as rings, modules, vector spaces, lattices, pseudocomplemented semilattice, MS algebras, and universal algebras (see [3][4][5][6][7][8]). More recently, fuzzy ideals of a poset were studied by Alaba et al. [9] as a generalization of those fuzzy ideals of lattices. However, Swamy and Raju [10,11] have unified all these fuzzy algebraic notions, by introducing the general theory of algebraic fuzzy systems. Some applications of fuzzy algebras and fuzzy points are also given in the literature [12][13][14].
It was Gougen [15] who first realized that the unit interval [0, 1] is insufficient to take truth values for fuzzy statements. Swamy and Swamy [16] have suggested that complete lattices satisfying the infinite meet distributive property are the most suitable candidates to take truth values of the general fuzzy statements.
On the other hand, in 1966, Diego [17] introduced the notion of Hilbert algebras and deductive systems and provided various properties. e theory of Hilbert algebras and deductive systems was further developed by Busneag in a series of papers [18,19]. Jun and Hong introduced the notion fuzzy deductive system in a Hilbert algebra in [20], and Jun constructed its extension in [21] whose truth values are in the unit interval [0, 1] of real numbers.
Initiated by all the above results, in this paper, we define fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law and we obtain several equivalent conditions for a fuzzy set to be a fuzzy deductive system. Fuzzy deductive systems generated by a fuzzy subset are characterized in different ways. It is also proved that the set of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system with respect to the pointwise ordering. Furthermore, we study fuzzy congruence relations on Hilbert algebras and we obtain a lattice isomorphism between the set of all fuzzy deductive systems and the set of all fuzzy congruence relations of a Hilbert algebra.

Preliminaries
In this section, we recall some definitions and basic properties of deductive systems of a Hilbert algebra. Definition 1. An algebra (A, * , 1) of type (2, 0) is said to be a Hilbert algebra if it satisfies the following conditions: for all x, y, z ∈ H and 1 is a fixed element of H.
It was proved by Diego in [17] that the class K of Hilbert algebras forms a variety. e following result is also adopted from [17]. Lemma 1. Every Hilbert algebra A satisfies the following conditions: { }, then A is called trivial. e binary relation ‵ ≤ ′ defined in a Hilbert algebra A by x ≤ y if and only if x * y � 1 is a partial order on A with the greatest element 1. is order is called the natural ordering on A.
The concept of a deductive system on a Hilbert algebra A was introduced by Diego [17] as follows.

Definition 2.
A nonempty subset D of a Hilbert algebra A is called a deductive system on A if it satisfies the following conditions: In other words, a deductive system of A is a subset of A containing 1 and closed under a deduction. One can easily check that the lattice of deductive systems of a Hilbert algebra A is closed under arbitrary intersection so that for any subset S of A, always there exists a smallest deductive system D(S) of A containing S. It is called the deductive system of A generated by S. Chajda [22] has characterized D(S) in the following way. Define sets for each positive integer n ≥ 2, inductively. en, where N denotes the set of positive integers. e internal structure of D(S) is also characterized by Diego in [17] as follows: roughout this paper, L stands for a complete lattice satisfying the infinite meet distributive law and A stands for a Hilbert algebra unless and otherwise stated.
By an L-fuzzy subset μ of A, we mean a mapping μ: A ⟶ L. e set μ(x): x ∈ A is called the image of μ and is denoted by Im(μ). An L-fuzzy subset μ of A is called normalized or unitary if 1 ∈ Im(μ). e class of all L-fuzzy subsets of A is denoted by L A .
For each α ∈ L, the α-level subset of μ, which is denoted by μ α , is a subset of A given by For any L-fuzzy subset μ of A and each x ∈ A, it was proved in [23] that For L-fuzzy subsets μ and σ of A, we write μ ⊆ σ to mean μ(x) ≤ σ(x) for all x ∈ A in the ordering of L. It can be easily verified that " ⊆ " is a partial order on the set L A and it is called the pointwise ordering.
For each x in A and 0 ≠ α in L, we define x α ∈ L A as follows: for each y ∈ A and we call it an L-fuzzy point of A.

L-Fuzzy Deductive Systems
In this section, we define L-fuzzy deductive system in a Hilbert algebra A and investigate some of its properties.
Definition 3. An L-fuzzy subset μ of A is said to be an L-fuzzy deductive system on A if We denote by FD L (A) the set of all L-fuzzy deductive systems of A. e following lemma is a straightforward verification of the definition, but useful to prove results in the latter section.
Corollary 1. Let S be a subset of A and 1 ≠ α ∈ L. en, S is a deductive system if and only if α S is an L-fuzzy deductive system of A.
Proof. Suppose that S is a deductive system of A and let α ∈ L − 1 { }. en, 1 ∈ S and hence, α Assume that α S (x) ≠ α and α S (x * y) ≠ α. en, x, x * y ∈ S. By our assumption, S is a deductive system of A so that y ∈ S and hence, us, α S is an L-fuzzy deductive system of A. Conversely, suppose that α S is an L-fuzzy deductive system of A for some e L-fuzzy deductive system α S given above is called the α-level L-fuzzy deductive system of A corresponding to the deductive system S.
Let us now define a binary operation ⊙ , called a product, on L A as follows: for any η, In the following theorem, we use this product and give a Rosenfeld-type characterization for L-fuzzy subsets of A to be an L-fuzzy deductive system.

Theorem 1. μ ∈ L A is an L-fuzzy deductive system of A if and only if μ(1) � 1 and for any normalized η ∈ L A and any
Proof. Suppose that μ is an L-fuzzy deductive system of A. Let η be a normalized L-fuzzy subset of A and σ is an L-fuzzy subset of A such that η ⊆ μ and η ⊙ σ ⊆ μ. en, η(a) � 1 for some a ∈ A. Now, for any x ∈ A, erefore, σ ⊆ μ. Conversely, suppose that μ ∈ L A satisfies the given condition. Now, we show that μ is an L-fuzzy deductive system of A. By hypothesis, μ(1) � 1. Let x, y ∈ A. Define L-fuzzy subsets η and σ of A by for any z ∈ A,

Journal of Mathematics
Clearly η is a normalized and η ⊆ μ. Moreover, for each z ∈ A, which is a subset of μ. Hence, by hypothesis, we have σ ⊆ μ.
e following result is another characterization for L-fuzzy deductive systems of A.
Conversely, suppose that μ satisfies the given conditions.
us, by hypothesis, we have μ(y) ≥ μ(x) ∧ μ(x * y). erefore, μ is an L-fuzzy deductive system of A. Proof. Let μ i : i ∈ Δ be any family of L-fuzzy deductive systems of A. Now, we claim that ∩ i∈Δ μ i is an L-fuzzy deductive system of A. If Δ � ∅, then by logical convention, for all x ∈ A, and clearly, it is an L-fuzzy deductive system of erefore, ∩ i∈Δ μ i is an L-fuzzy deductive systems of A. is lemma confirms that the class FD L (A) of all L-fuzzy deductive systems of A forms a closure fuzzy set system and for any L-fuzzy subset μ of A, always there exists a smallest L-fuzzy deductive system of A containing μ which we call it the L-fuzzy deductive system of A generated by μ and is denoted by D L (μ). □ Lemma 5. Let μ and η be L-fuzzy subsets of A. en, It can be deduced from this lemma that the map μ ↦ D L (μ) forms a closure operator, i.e., isotone, extensive and idempotent on the lattice L A . Moreover, L-fuzzy deductive systems of A are those closed elements of L A with respect to this closure operator. In the following theorem, we characterize L-fuzzy deductive systems generated by an L-fuzzy subset in terms of their level sets.

Theorem 3. For an L-fuzzy subset λ of A, let λ be an L-fuzzy
for all x ∈ A. en, λ � D L (λ).
Proof. We show that λ is the smallest L-fuzzy deductive system of A containing λ. Let us first show that λ is an L-fuzzy deductive system: (1) Let x, y ∈ A. en, consider 4

Journal of Mathematics
If we put c � α ∧ β, then we obtain λ α ⊆ λ c and λ β ⊆ λ c , which gives the following: erefore, x, x * y ∈ D(λ c ) and hence y ∈ D(λ c ). Now it follows from the above-given equality that erefore, λ is an L-fuzzy deductive system of A. It is also clear to see that λ ⊆ λ. Suppose that μ is any other L-fuzzy deductive system of A such that λ ⊆ μ. en, it is clear that erefore, λ is the smallest L-fuzzy deductive system containing λ and hence λ � D L (λ).
In the following theorem, we characterize L-fuzzy deductive systems generated by an L-fuzzy subset using finitely generated crisp deductive systems.

Proof. By eorem 3, it is enough if we show that λ � λ.
For each x ∈ A, let us define two sets H x and G x as follows: Our claim is to show that In one way, we show that H x ⊆ G x . If α ∈ H x , then α � ∧ a∈F λ(a) and x ∈ D(F) for some finite subset F of A. at is, a ∈ λ α , for all a ∈ F and x ∈ D(F). Hence, x ∈ D(λ α ).
en, α ∈ G x and hence H x ⊆ G x . In the other way, we prove that, for each α ∈ G x , there exists β ∈ H x such that α ≤ β. Let α ∈ G x .
en, x ∈ D(λ α ); that is, there exist x 1 , x 2 , . . . , x n ∈ λ α such that x 1 * (x 2 * (· · · (x n * x) · · ·)) � 1. is implies that ∧ n i�1 λ(a i ) ≥ α. Moreover, if we put β � ∧ n i�1 λ(x i ) and F � x 1 , x 2 , . . . , x n , then F is a finite subset of A such that x ∈ D(F). us, β ∈ H x such that α ≤ β. □ Corollary 3. For each x ∈ A and α ∈ L − 0 { }, the L-fuzzy deductive system of A generated by the fuzzy point x α is characterized as The following theorem gives a description for α−level cuts of L-fuzzy deductive systems generated by an L-fuzzy subset.
is means that x ∈ ∩ c∈M D(μ c ) and α ≤ supM. us, x ∈ H, and hence, the proof ends.

. Moreover, if L is a chain and μ is finite valued or equivalently if μ has sup property, then the equality holds.
In the following, we give an algebraic characterization for L-fuzzy deductive systems generated by L-fuzzy subset. Let μ be an L-fuzzy subset of A and N be a set of positive integers. For each n ∈ N, let us define L-fuzzy subsets μ n of A as follows: for all x ∈ A, and for each n ≥ 1 and each y ∈ A, Then, it is immediate from the definition that μ n (1) � 1 for all n ≥ 1. Moreover, one can deduce that
Proof. Put μ � ⋃ μ n : n ∈ N . Now, we claim that μ is the smallest L-fuzzy deductive system containing μ. Now, let x ∈ A. en, So μ is an L-fuzzy deductive system of A. Again, let θ be any L-fuzzy deductive system of A such that μ ⊆ θ. We show that μ n ⊆ θ for all n ∈ N. We use induction on n. Let x ∈ A. en, if x � 1, then θ(1) � 1 � μ 1 (x). If x ≠ 1, then θ(x) ≥ μ(x) � μ 1 (x). is implies that μ 1 ⊆ θ and hence it holds for n � 1. Let n > 1 and assume the result to be true for some n − 1, i.e., μ n−1 ⊆ θ. Now, for any x ∈ A, we have Hence, μ n ⊆ θ. us, by mathematical induction, we have μ n ⊆ θ for all n ∈ N. Hence, μ � ⋃ μ n : n ∈ N ⊆ θ. erefore, μ � D L (μ). e following is also another algebraic characterization of L-fuzzy deductive system generated by an L-fuzzy subset of A.

Theorem 7. Let λ be an L-fuzzy subset of A.
en, the L-fuzzy subset λ defined by: λ(1) � 1 and for x ≠ 1, is the L-fuzzy deductive system of A generated by λ.
The following theorem gives a method of constructing L-fuzzy deductive systems using crisp deductive systems in the sense of [10].

Theorem 8. Suppose that H α α∈L is a family of deductive systems of
for all M ⊆ L. en, there is a unique L-fuzzy deductive system μ of A for which μ α � H α for all α ∈ L. Moreover, every L-fuzzy deductive system of A is obtained in this way only.
Proof. We first show that the map α ↦ H α is antitone, in the sense that, for each α, β ∈ L, for all x ∈ A. Clearly, μ is well defined. Our aim is to show that μ α � H α for all α ∈ L. e inclusion H α ⊆ μ α follows easily from the definition of μ. To prove the other inclusion, let x ∈ μ α . en, μ(x) ≥ α, i.e., If we put M � c ∈ L: x ∈ H c , then M ⊆ L such that α ≤ supM and x ∈ H c for all c ∈ M, i.e., By our assumption, it follows that x ∈ H supM . Since α ≤ supM and the map α ↦ H α is antitone, we obtain x ∈ H α . us, H α � μ α . is means that μ is an L-fuzzy subset of A for which its α-level sets are H α 's. Each H α being a deductive system of A, it follows from Lemma 3 that μ is an L-fuzzy deductive system. e uniqueness of μ follows from the fact μ α � H α for all α ∈ L. e converse is straightforward.

Lattice of L-Fuzzy Deductive Systems
Let us first recall some important results from [24]. Let C be a nonempty collection of fuzzy subsets of a nonempty set X.
Definition 5. C is said to be a closure system in L X if it is closed under arbitrary intersection of fuzzy sets; i.e., if for any subcollection D of fuzzy subsets of X in C, it holds that (43) A closure system of fuzzy sets is also known as the "Moor family" of fuzzy sets.

Remark 2.
If C is a closure system in L X , then it contains the fuzzy subset 1 X of X. is is because the fuzzy set 1 X can be expressed as the infimum of an empty collection of fuzzy subsets of X. Theorem 9. If C is a closure system in L X , then (C, ⊆ ) forms a complete lattice, where ⊆ is the inclusion ordering of fuzzy sets. Definition 6. A nonempty collection C of fuzzy subsets of X is called inductive if every nonempty chain in C has a supremum in C.

Definition 7.
A closure system C in L X is said to be an algebraic closure fuzzy sets system if it is inductive.
As observed in the previous section, the collection FD L (A) of all L-fuzzy deductive systems of A is closed under arbitrary intersection of fuzzy sets.
us, (FD L (A), ⊆ ) forms a closure fuzzy set system and hence it is a complete lattice. e following theorem summarizes this.

Theorem 10.
e collection FD L (A) forms a complete lattice with respect to the pointwise ordering, where the infimum and supremum of any of its subfamily μ i : i ∈ Δ , respectively, are We show that η is an L-fuzzy deductive system of A. Clearly, η(1) � 1. Let x, y ∈ A. First observe that, for each i, j ∈ Δ, there exists k ∈ i, j such that μ i ⊆ μ K and μ j ⊆ μ k . Now, consider the following: erefore η is an L-fuzzy deductive system of A. □ Remark 3. If L is an algebraic lattice, then FC a (A) forms an algebraic lattice.

L-Fuzzy Congruences
By an L-fuzzy relation on A, we mean an L-fuzzy subset of A × A. For any α ∈ L and an L-fuzzy relation Θ on A, the set is called the α-level relation of Θ on A.
A reflexive, symmetric, and transitive L-fuzzy relation on A is called an L-fuzzy equivalence relation on A.
for all a, b, c, d ∈ A. is fact is often expressed in the way that Θ is said to have the substitution property with respect to the binary operation * .

Definition 10. A compatible L-fuzzy equivalence relation on
A is called an L-fuzzy congruence relation on A.
We denote by FCon(A) the class of all L-fuzzy congruence relations on A, and it is clear that FCon(A) is a complete lattice. For any a ∈ L and Θ ∈ FCon(A), define a fuzzy subset [a] Θ of A by for all x ∈ A. We call [a] Θ an L-fuzzy congruence class of Θ determined by a, and in particular, [1] Θ is called the kernel of Θ. Let us put and define a binary operation ⊛ on A/Θ by en, it is routine to verify that ((A/Θ), ⊛, [1] Θ ) is a Hilbert algebra and it is called the quotient Hilbert algebra of A modulo Θ.
(52) us, [1] Θ is an L-fuzzy deductive system of A. □ Remark 4. It is sufficient for Θ to be reflexive and compatible only to obtain the result of Lemma 7.

Lemma 8. For each Θ, Φ ∈ FCon(A),
and put Θ(x, y) � α. en, it is clear that Θ α is a congruence relation on A containing (x, y) and hence (x * y, 1) and [1] and so the following hold in the quotient Hilbert algebra (A/Φ α ): (A/Φ α )Since the binary operations x * y and y * x form a G€ odel equivalence system in the variety of Hilbert algebras, it follows that Φ α [x] � Φ α [y]. erefore, (x, y) ∈ Φ α and thus, Φ(x, y) ≥ α � Θ(x, y). Similarly, we can show that Θ(x, y) ≥ Φ(x, y) so that the equality holds. □ Corollary 6. For each Θ, Φ ∈ FCon(A), [1] for all α ∈ L. Let x, y ∈ A, and put Θ(x, y) � α. en, it is clear that Θ α is a congruence relation on A containing (x, y) and hence (x * y, 1) and (y * x, 1) belong to Θ α . is imply that x * y, y * x ∈ Θ α [1] ⊆ Φ α [1] and so the following hold in the quotient Hilbert algebra (A/Φ α ): (A/Φ α )Since the binary operations x * y and y * x form a G€ odel equivalence system in the variety of Hilbert algebras, it follows that For any L-fuzzy subset μ of A, let us define an L-fuzzy relation Θ μ of A as follows: for all a, b ∈ A.

Lemma 9.
If the fuzzy relation Θ μ is reflexive, then the kernel of Θ μ is μ.
Proof. Let Θ μ be reflexive. en, for any x ∈ A, us, μ(1) � 1. Now, since for any x ∈ A, (58) erefore, the kernel of Θ μ is μ. □ Lemma 10. If Θ μ be an L-fuzzy equivalence relation on A, then for any a, b ∈ A with a ≤ b, it holds that Proof. Let Θ μ be an L-fuzzy equivalence relation on A and a, b ∈ A such that a ≤ b. en, a * b � 1. By Lemma 9, we have μ(1) � 1 � μ(a * b) and [1] Θ μ � μ. us, is proves the result. □ Theorem 11. If μ is an L-fuzzy deductive system of A, then Θ μ is an L-fuzzy congruence relation on A whose kernel is μ.

Corollary 8. e map Θ↦[1] Θ is a lattice isomorphism between FCon(A) and FD L (A).
Proof. Lemma 8 confirms that the map is one-one. Moreover, for any μ ∈ FD L (A), it is proved in the above theorem that Θ μ ∈ FCon(A) such that [1] en, the map Θ⟼[1] Θ is onto. Furthermore, it is proved in Corollary 7 that this map is a lattice homomorphism. erefore, it is a lattice isomorphism between FCon(A) and FD L (A).

Data Availability
No data were used to support the results of this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.