Fuzzy Spectral Spaces and Fuzzy Congruences of a Heyting ADL
In this paper, the space of fuzzy prime ideals of Heyting almost distributive lattice is studied, and it is shown that the collection of all sets is a topology on , where is a fuzzy ideal on and . The only compact subset of the space is given. A fuzzy congruence relation on is defined, and the homomorphism between the set of all fuzzy ideals of and the set of all fuzzy ideals of is established. Furthermore, we established an isomorphism between fuzzy spectrum of and fuzzy spectrum of .
The class of distributive lattices has many interesting properties, which lattices, in general, do not have. For this reason, Swamy and Rao  introduced the concept of an almost distributive lattice (ADL) as a common abstraction of lattice and ring theoretic generalizations of a Boolean algebra. In , it is proved that the commutativity of , the commutativity of , the right distributivity of over , and the absorption law are all equivalent to each other and whenever any one of these properties holds, an ADL becomes a distributive lattice. The concept of an ideal in was introduced analogous to that in a distributive lattice, and it was observed that the set of all principal ideals of A forms a distributive lattice.
These results enable us to extend many existing concepts from the class of distributive lattices to the class of ADLs. Rao et al.  introduced the concept of Heyting almost distributive lattices (HADLs) as a generalization of Heyting algebra in the class of ADLs, and HADL is characterized in terms of the set of all its principal ideals. Rao and Ravi Kumar  proved that given any multiplicatively closed subset of disjoint from an ideal of , there exists a prime ideal belonging to which is disjoint from .
Sambasiva Rao and Rao in  studied topological properties of the space of prime ideals of a HADL. These authors proved that there is an isomorphism between a Heyting algebra and the open base of its prime spectrum. This mapping is epimorphism, and it is isomorphism if and only if the HADL is a Heyting algebra.
In , the concept of fuzzy set theory as a generalization of classical set theory was introduced by Zadeh. Rosenfield  started the pioneering work in the domain of fuzzification of algebraic objects on fuzzy groups. In particular, Bo and Wangming  and Santhi Sundar Raj et al.  have laid down the foundation for fuzzy ideals of a lattice and an ADL, respectively. In [9, 10], Alemayehu et al. introduced the concepts of belligerent fuzzy GE-filter of GE-algebras and beta-fuzzy filters of stone almost distributive lattices, respectively. Santhi Sundra Raj et al.  introduced the concept of fuzzy prime ideals of almost distributive lattices. In 1998, Swamy and Raju  introduced the concept of fuzzy ideals and fuzzy congruences of distributive lattices and showed that there is a one-to-one correspondence between the lattice of fuzzy ideals and the lattice of fuzzy congruences of . In , Alaba and Addis studied the fuzzy congruence relations of an ADL , and they gave the smallest fuzzy congruence on such that its quotient is a distributive lattice.
This paper comprises of four sections. The first two sections deal with the introductory and preliminary concepts. In Section 3 of this paper, we studied some properties of fuzzy prime ideals of a HADL , and it is shown that the collection is a topology on . The fact that the set is the only compact subset of the space is proved. Furthermore, it is shown that an ADL is a distributive lattice if and only if the mapping between and where is injective. In Section 4, a fuzzy congruence relation on a HADL is defined, and the homomorphism between the set of all fuzzy ideals of and the set of all fuzzy ideals of is established, where . Furthermore, we proved that the mapping between fuzzy spectrum of and fuzzy spectrum of is bijective.
In this section, we recall basic definitions and results which will be used in this article.
Definition 1. An algebra of type is called an almost distributive lattice if it satisfies the following conditions: for all and ,(1)(2)(3)(4)(5)(6)
Let and “ ” be a partial ordering on . We read is less than or equal to and write if . Equivalently, . If an element is a maximal with respect to the partial ordering on , then is said to be maximal.
Theorem 1. Let be an ADL and. Then, the following are equivalent:(1) is maximal with respect to(2), for all(3), for all
Definition 2. Let be an ADL with 0 and a maximal element . Suppose is a binary operation on satisfying the following conditions:(1)(2)(3)(4)(5) for all Then, is called a Heyting almost distributive lattice or simply a HADL.
Lemma 1. Let be a HADL. Then, for, the following hold:(1)(2)(3)
Definition 3. Let and be two HADLs. Then, the mapping is called a homomorphism of into if for any , the following conditions hold:(1)(2)(3)(4)
For any set , a function is called a fuzzy subset of , where is a unit interval of real numbers.
Definition 4. Let be a fuzzy subset of an ADL . For any , we denote the level subset , i.e.,
Definition 5. A fuzzy subset of an ADL is said to be a fuzzy ideal of if and only if(1)(2) for all
From , remember that for each and , the fuzzy subset of given byfor all is called a fuzzy point of . In this case, an element of is called the support of and is its value. For a fuzzy point of and a fuzzy subset of , we write to say that .
3. Fuzzy Prime Spectrum of HADLs
This section deals with some properties of fuzzy prime ideals of HADL. The fact that the set (where is the class of all fuzzy prime ideals of a HADL ) is the only compact subset of the space is proved. Furthermore, it is shown that an ADL is a distributive lattice if and only if the mapping between and where is injective. Unless otherwise mentioned stands for Heyting almost distributive lattices.
Definition 6. A fuzzy subset of is called a fuzzy ideal if for all , it satisfies the following conditions:(1)(2)(3)
From this definition, we have got that is a fuzzy ideal of if and only if one of the following conditions holds:(1) and for all (2) and and , for all
Definition 7. A fuzzy subset of is called a fuzzy filter if for all and a maximal element , it satisfies the following conditions:(1)(2)(3)
From this definition, we have got that is a fuzzy filter of if and only if one of the following conditions holds:(1) and for all (2) and and , for all
Example 1. Let. Define the binary operations,, and on as follows:
Then, is a Heyting ADL with maximal elements and .
For fuzzy subsets and of , define , and , and , , and . Then, it can be easily checked that is a fuzzy ideal of and is a fuzzy filter of .
Theorem 2. Let be a fuzzy ideal in and. Then, .
Proof. Suppose that is a fuzzy ideal of and . Then, .
With similar procedure, . Hence, .
Theorem 3. Let be a fuzzy filter in and. Then, .
Remark 1. In Example 1, we have, , , and.
We denote the class of all fuzzy ideals of by .
Theorem 4. The set forms a Heyting algebra.
Proof. Clearly, is a bounded distributive lattice with greatest element and least element . In addition to this, the intersection of a class of fuzzy ideals is a fuzzy ideal. This implies that it forms a complete lattice. For any two fuzzy ideals , define an operation on as follows:Let and be fuzzy ideals of such that . Now we prove that is the largest fuzzy ideal of . Since and , we have got that . Again for any , For each such that and . Then, . Thus, . Now we show that and . Thus, . Similarly, . So, . Hence, is a fuzzy ideal of . From the definition of and the given condition , it is clear that . Suppose that is a fuzzy ideal of containing such that . It contradicts the definition of . Hence, is the largest fuzzy ideal of such that for any .
Therefore, is a Heyting algebra.
Lemma 2. Let be a fuzzy ideal of,, and. If, then there exists a prime fuzzy ideal of such that and.
Let denote the set of all fuzzy prime ideals of . For any fuzzy ideal of , define , and for any fuzzy point , define . Then, we have the following results.
Theorem 5. Let be a fuzzy ideal of. Then, if and only if.
Proof. Let and . Then, . Hence, we get . Thus, . So, . Conversely, assume that . Suppose that . This implies that . Then, from Lemma 2, there exists such that and . This shows that so that . It follows that where which is a contradiction. Therefore, .
Lemma 3. For any fuzzy ideals and of, the following conditions hold:(1) if and only if(2)(3)(4)
Proof. (1)Suppose and are fuzzy ideals of such that . Let . This implies that . This implies that , and so . Hence, . Conversely, suppose that . We prove that . Assume . This implies that for some . Put . This implies that and . By Theorem 5, we have and . Hence, , which is contradiction. Hence, .(2)Clearly, . Conversely, let . Suppose that . Then, and . This means and where . Consequently, . Therefore, .(3)Clearly, . On the other hand, if , then . Suppose that and . This shows that which is a contradiction. Thus, or . Following this, or where . It follows that . Hence, .(4)Let . Then, . This implies that there exists such that and . This implies that for some .Conversely, for some fuzzy ideals of . Let be such that .Thus, .Hence, .
Next, we discus the space of fuzzy prime ideals of HADL analogous with the hull-kernel topology.
Theorem 6. Let is a fuzzy ideal of. Then, is a topology on.
Proof. Consider the fuzzy subsets of defined as and for all . Clearly, and are fuzzy ideals of . for all , which is impossible. Thus, . Since each is nonconstant, for all . Thus, . This implies .
Also, for any fuzzy ideals and of , by (2) in Lemma 3, we have . This shows that is closed under finite intersections. Next, let be any family of fuzzy ideals of . Now we prove that . Let ; then, , which implies that for some . Otherwise, if for each , it will be true that . Thus, where . Clearly, . Hence, . Therefore, is closed under arbitrary unions, and hence it is topology on .
The topological space is called the fuzzy prime spectrum of , and it is denoted by .
Theorem 7. Let and be fuzzy subsets of. Then, the following conditions hold:(1)(2)
Corollary 1. For any and, the following conditions hold:(1)If, then(2)(3)(4)(5)(6)(7)(8)
From Corollary 1, we have the following.
Corollary 2. Let. Then, forms a base for a topology on.
Corollary 3. is a compact space.
For any fuzzy subset of , is open set of and is a closed set of . Also, every closed set in is of the form for all fuzzy subsets of . Then, we have the following.
Theorem 8. Let. Then, the closure of (i.e., ) is given by.
Theorem 9. Let