#### Abstract

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.

#### 1. Introduction

The pioneering work of Zadeh [1] on fuzzy subsets of a set has been extensively applied to many scientific fields. This 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, pseudo-complemented semilattice, MS algebras, and universal algebras (see [3–8]). 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 [12–14].

It was Gougen [15] who first realized that the unit interval 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. The 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 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.

#### 2. Preliminaries

In this section, we recall some definitions and basic properties of deductive systems of a Hilbert algebra.

*Definition 1. *An algebra of type is said to be a Hilbert algebra if it satisfies the following conditions:(1)(2)(3)for all and is a fixed element of .

It was proved by Diego in [17] that the class of Hilbert algebras forms a variety. The following result is also adopted from [17].

Lemma 1. *Every Hilbert algebra satisfies the following conditions:*(1)*(2)**(3)**(4)**(5)**(6)**(7)*

If , then A is called trivial. The binary relation defined in a Hilbert algebra by if and only if is a partial order on *A* with the greatest element . This order is called the natural ordering on *A*.

The concept of a deductive system on a Hilbert algebra was introduced by Diego [17] as follows.

*Definition 2. *A nonempty subset of a Hilbert algebra is called a deductive system on if it satisfies the following conditions:(1)(2)(3)In other words, a deductive system of is a subset of containing and closed under a deduction. One can easily check that the lattice of deductive systems of a Hilbert algebra is closed under arbitrary intersection so that for any subset of , always there exists a smallest deductive system of containing . It is called the deductive system of generated by . Chajda [22] has characterized in the following way. Define setsfor each positive integer , inductively. Then,where denotes the set of positive integers.

The internal structure of is also characterized by Diego in [17] as follows: if and for ,Throughout this paper, stands for a complete lattice satisfying the infinite meet distributive law and stands for a Hilbert algebra unless and otherwise stated.

By an -fuzzy subset of , we mean a mapping . The set is called the image of and is denoted by . An -fuzzy subset of is called normalized or unitary if . The class of all -fuzzy subsets of is denoted by .

For each , the -level subset of , which is denoted by , is a subset of given byFor any -fuzzy subset of and each , it was proved in [23] thatFor -fuzzy subsets and of , we write to mean for all in the ordering of . It can be easily verified that “” is a partial order on the set and it is called the *pointwise ordering*.

For each in and in *L*, we define as follows:for each and we call it an -fuzzy point of .

#### 3. *L*-Fuzzy Deductive Systems

In this section, we define -fuzzy deductive system in a Hilbert algebra and investigate some of its properties.

*Definition 3. *An -fuzzy subset of is said to be an -fuzzy deductive system on if(1)(2)We denote by the set of all -fuzzy deductive systems of .

The following lemma is a straightforward verification of the definition, but useful to prove results in the latter section.

Lemma 2. *Let be an -fuzzy deductive system on . Then, for any , the following hold:*(1)*(2)**(3)**(4)*

*Proof. *(1)Let such that . Then, we have the following:(2)Let such that . Then, and hence, . Thus, it follows from (1) that .(3)Let such that . Then, . By (2), we have . This implies that(4)Let . By Lemma 1, . Then, it follows from (3) that .

Lemma 3. *An -fuzzy subset of is an -fuzzy deductive system of if and only if the -level subset is a deductive system of , for all .*

*Proof. *Let be an -fuzzy deductive system of and . Then, Thus, . Again, let such that and . Then,This implies that and hence . So is a deductive system of .

Conversely, suppose that is a deductive system of for all . In particular, is a deductive system. Since , we have , but , and hence . Thus, we have . Again, let and put . This implies that and so we have . Therefore,Hence, is an -fuzzy deductive system of .

*Definition 4. *For any subset of and each , define an -fuzzy subset of by for any

Corollary 1. *Let be a subset of and . Then, is a deductive system if and only if is an -fuzzy deductive system of .*

*Proof. *Suppose that is a deductive system of and let . Then, and hence, . Let . If either or , thenAssume that and . Then, . By our assumption, is a deductive system of so that and hence,Thus, is an -fuzzy deductive system of . Conversely, suppose that is an -fuzzy deductive system of for some . Then, and hence, . Also, let such that . Then, , which gives . being an -fuzzy deductive system, we obtain . Equivalently, and hence is a deductive system of . This completes the proof.

*Remark 1. *The -fuzzy deductive system given above is called the -level -fuzzy deductive system of corresponding to the deductive system .

Let us now define a binary operation , called a product, on as follows: for any , for all .

In the following theorem, we use this product and give a Rosenfeld-type characterization for -fuzzy subsets of to be an -fuzzy deductive system.

Theorem 1. * is an -fuzzy deductive system of if and only if and for any normalized and any , and together imply .*

*Proof. *Suppose that is an -fuzzy deductive system of . Let be a normalized -fuzzy subset of and is an -fuzzy subset of such that and Then, for some . Now, for any ,Therefore, .

Conversely, suppose that satisfies the given condition. Now, we show that is an -fuzzy deductive system of . By hypothesis, . Let . Define -fuzzy subsets and of by for any ,Clearly is a normalized and . Moreover, for each ,which is a subset of . Hence, by hypothesis, we have . Hence, and so . Therefore, is an -fuzzy deductive system.

*Notation 1. *We write , to say that is a finite subset of . The following result is another characterization for -fuzzy deductive systems of .

Theorem 2. *An -fuzzy subset of is an -fuzzy deductive system of if and only if for any , for all .*

*Proof. *Suppose that is an -fuzzy deductive system of . Let and put . Then, for all and hence . Clearly, by Lemma 3, is a deductive system. Therefore, and hence, for all .

Conversely, suppose that satisfies the given conditions. Now, since , by hypothesis, we have and hence .

Let . Put . Then, it is easy to see that . Thus, by hypothesis, we have . Therefore, is an -fuzzy deductive system of .

#### 4. *L*-Fuzzy Deductive Systems Generated by *L*-Fuzzy Subset

Lemma 4. *The class is closed under arbitrary intersection.*

*Proof. *Let be any family of -fuzzy deductive systems of . Now, we claim that is an -fuzzy deductive system of . If , then by logical convention, , where is an -fuzzy subset of defined byfor all , and clearly, it is an -fuzzy deductive system of . Assume that . Since , for all , we have . Again, let . Then,Therefore, is an -fuzzy deductive systems of .

This lemma confirms that the class of all -fuzzy deductive systems of forms a closure fuzzy set system and for any -fuzzy subset of , always there exists a smallest -fuzzy deductive system of containing which we call it the -fuzzy deductive system of generated by and is denoted by .

Lemma 5. *Let and be -fuzzy subsets of . Then,*(1)*(2)**(3)**It can be deduced from this lemma that the map forms a closure operator, i.e., isotone, extensive and idempotent on the lattice . Moreover, -fuzzy deductive systems of are those closed elements of with respect to this closure operator.*

Lemma 6. *For any subset of and each , .*

*Proof. *We show that is the smallest -fuzzy deductive system of containing . Since is a deductive system of containing , it is clear that is an -fuzzy deductive system of containing . Suppose that is an -fuzzy deductive system of such that . Then, for all . Let be any element in . If , then . Let . Since , we have . This implies that . So and hence . Therefore, so that . Therefore, .

Corollary 2. *Let be any subset of and its characteristic function. Then, .**In the following theorem, we characterize -fuzzy deductive systems generated by an -fuzzy subset in terms of their level sets.*

Theorem 3. *For an -fuzzy subset of , let be an -fuzzy subset of defined byfor all . Then, .*

*Proof. *We show that is the smallest -fuzzy deductive system of containing . Let us first show that is an -fuzzy deductive system:(1)(1)Let . Then, considerIf we put , then we obtain and , which gives the following:Therefore, and hence . Now it follows from the above-given equality thatTherefore, is an -fuzzy deductive system of . It is also clear to see that . Suppose that is any other -fuzzy deductive system of such that . Then, it is clear that for all . Now, for any , considerTherefore, is the smallest -fuzzy deductive system containing and hence .

In the following theorem, we characterize -fuzzy deductive systems generated by an -fuzzy subset using finitely generated crisp deductive systems.

Theorem 4. *For an -fuzzy subset of , let be an -fuzzy subset of defined byfor all . Then, .*

*Proof. *By Theorem 3, it is enough if we show that . For each , let us define two sets and as follows:Our claim is to show thatIn one way, we show that . If , then and for some finite subset of . That is, , for all and . Hence, . Then, and hence . In the other way, we prove that, for each , there exists such that . Let . Then, ; that is, there exist such that . This implies that .

Moreover, if we put and , then is a finite subset of such that . Thus, such that .

Corollary 3. *For each and , the -fuzzy deductive system of generated by the fuzzy point is characterized asfor all .**The following theorem gives a description for level cuts of -fuzzy deductive systems generated by an -fuzzy subset.*

Theorem 5. *Let be an -fuzzy subset of and :*

*Proof. *Let us putIf , then for some with ; i.e., for all and . By Theorem 3, we have the following:Therefore, for all . This gives . Thus, and hence . To prove the other inequality, let us take . Then,If we put , then such that and for all . This means that and . Thus, , and hence, the proof ends.

Corollary 4. *For any -fuzzy subset of and each , . Moreover, if is a chain and is finite valued or equivalently if has sup property, then the equality holds.**In the following, we give an algebraic characterization for -fuzzy deductive systems generated by -fuzzy subset. Let be an -fuzzy subset of and be a set of positive integers. For each , let us define -fuzzy subsets of as follows:for all , and for each and each ,**Then, it is immediate from the definition that for all . Moreover, one can deduce that*

Theorem 6. *Let be an -fuzzy subset of . Then, .*

*Proof. *Put . Now, we claim that is the smallest -fuzzy deductive system containing . Now, let . Then,Thus, we have .

Since , we have . Let and .

ThenSo is an -fuzzy deductive system of . Again, let be any -fuzzy deductive system of such that . We show that for all . We use induction on . Let . Then, if , then . If , then . This implies that and hence it holds for . Let and assume the result to be true for some , i.e., . Now, for any , we haveHence, . Thus, by mathematical induction, we have for all Hence, . Therefore, .

The following is also another algebraic characterization of -fuzzy deductive system generated by an -fuzzy subset of .

Theorem 7. *Let be an -fuzzy subset of . Then, the -fuzzy subset defined by: and for ,is the -fuzzy deductive system of generated by .**The following theorem gives a method of constructing -fuzzy deductive systems using crisp deductive systems in the sense of [10].*

Theorem 8. *Suppose that is a family of deductive systems of such thatfor all . Then, there is a unique -fuzzy deductive system of for which for all . Moreover, every -fuzzy deductive system of is obtained in this way only.*

*Proof. *We first show that the map is antitone, in the sense that, for each . Let such that . Put . Then, . By our hypothesis . Therefore, and hence, the map is antitone. Define an -fuzzy subset of byfor all . Clearly, is well defined. Our aim is to show that for all . The inclusion follows easily from the definition of . To prove the other inclusion, let . Then, , i.e.,If we put , then such that and for all , i.e.,By our assumption, it follows that . Since and the map is antitone, we obtain . Thus, . This means that is an -fuzzy subset of for which its -level sets are ’s. Each being a deductive system of , it follows from Lemma 3 that is an -fuzzy deductive system. The uniqueness of follows from the fact for all . The converse is straightforward.

#### 5. Lattice of *L*-Fuzzy Deductive Systems

Let us first recall some important results from [24]. Let be a nonempty collection of fuzzy subsets of a nonempty set .

*Definition 5. * is said to be a closure system in if it is closed under arbitrary intersection of fuzzy sets; i.e., if for any subcollection of fuzzy subsets of in , it holds thatA closure system of fuzzy sets is also known as the “Moor family” of fuzzy sets.

*Remark 2. *If is a closure system in , then it contains the fuzzy subset of . This is because the fuzzy set can be expressed as the infimum of an empty collection of fuzzy subsets of .

Theorem 9. *If is a closure system in , then forms a complete lattice, where is the inclusion ordering of fuzzy sets.*

*Definition 6. *A nonempty collection of fuzzy subsets of is called inductive if every nonempty chain in has a supremum in .

*Definition 7. *A closure system in is said to be an algebraic closure fuzzy sets system if it is inductive.

As observed in the previous section, the collection of all -fuzzy deductive systems of is closed under arbitrary intersection of fuzzy sets. Thus, forms a closure fuzzy set system and hence it is a complete lattice. The following theorem summarizes this.

Theorem 10. *The collection forms a complete lattice with respect to the pointwise ordering, where the infimum and supremum of any of its subfamily , respectively, are*

Corollary 5. *The set forms the algebraic closure fuzzy set system on with respect to the pointwise ordering of fuzzy sets.*

*Proof 13. *It is enough to show that is inductive in . Let be a chain in . Let us putWe show that is an -fuzzy deductive system of . Clearly, . Let . First observe that, for each , there exists such that and . Now, consider the following:Therefore is an -fuzzy deductive system of .

*Remark 3. *If is an algebraic lattice, then forms an algebraic lattice.

#### 6. *L*-Fuzzy Congruences

By an -fuzzy relation on , we mean an -fuzzy subset of . For any and an -fuzzy relation on , the setis called the -level relation of on .

*Definition 8. *An -fuzzy relation on is said to be(1)Reflexive if for all (2)Symmetric if for all (3)Transitive if for each , for all A reflexive, symmetric, and transitive -fuzzy relation on is called an -fuzzy equivalence relation on .

*Definition 9. *An -fuzzy relation on is said to be compatible, iffor all .

This fact is often expressed in the way that is said to have the substitution property with respect to the binary operation .

*Definition 10. *A compatible -fuzzy equivalence relation on is called an -fuzzy congruence relation on .

We denote by the class of all -fuzzy congruence relations on , and it is clear that is a complete lattice. For any and , define a fuzzy subset of byfor all . We call an -fuzzy congruence class of determined by , and in particular, is called the kernel of . Let us putand define a binary operation on byThen, it is routine to verify that is a Hilbert algebra and it is called the quotient Hilbert algebra of modulo .

Lemma 7. *For each , is an -fuzzy deductive system of .*

*Proof. *Since is reflexive, we have . Let . Then,Thus, is an -fuzzy deductive system of .

*Remark 4. *It is sufficient for to be reflexive and compatible only to obtain the result of Lemma 7.

Lemma 8. *For each ,*

*Proof. *Let such that . Then, it is clear that for all . Let , and put . Then, it is clear that is a congruence relation on containing and hence and belong to . This imply that and so the following hold in the quotient Hilbert algebra :Since the binary operations form a Gdel equivalence system in the variety of Hilbert algebras, it follows that . Therefore, and thus, . Similarly, we can show that so that the equality holds.

Corollary 6. *For each ,*

*Proof. *Let such that . Then, it is clear that for all . Let , and put . Then, it is clear that is a congruence relation on containing and hence and belong to . This imply that and so the following hold in the quotient Hilbert algebra :Since the binary operations form a Gdel equivalence system in the variety of Hilbert algebras, it follows that . Therefore, and hence . Thus .

Corollary 7. *For each ,*(1)*(2)**For any -fuzzy subset of , let us define an -fuzzy relation of as follows:for all .*

Lemma 9. *If the fuzzy relation is reflexive, then the kernel of is .*

*Proof. *Let be reflexive. Then, for any , . Thus, . Now, since for any ,Therefore, the kernel of is .

Lemma 10. *If be an -fuzzy equivalence relation on , then for any with , it holds that*

*Proof. *Let be an -fuzzy equivalence relation on and such that . Then, . By Lemma 9, we have and . Thus,