#### Abstract

The notion of an Almost Distributive Lattice (ADL) is a common abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebra and Boolean rings. In this paper, the set of all -fuzzy prime ideals of an ADL with truth values in a complete lattice satisfying the infinite meet distributive law is topologized and the resulting space is discussed.

#### 1. Introduction

The concept of prime ideal is vital in the study of structure theory of distributive lattices in general and of Boolean algebras in particular [2]. In this context, we recall the work of Stone [1] on the representation of distributive lattices by algebra of sets. In fact, he proved that a lattice is distributive if and only if any ideal of is the intersection of all prime ideals containing it. Also, he introduced a topology on the set of all prime ideals of a given Boolean algebra in such a way that is isomorphic with the Boolean algebra of cl-open subsets of resulting space.

Swamy and Rao [2] have introduced the notion of an Almost Distributive Lattice (ADL) which is algebra of type satisfying all the axioms of a distributive lattice with zero except commutative, commutative, and right distributivity of over . In fact, in any ADL, three conditions are equivalent.

Next, Rosenfold [3] introduced the notion of fuzzy groups; many researchers are turned into fuzzifying various algebra. Santhi Sundar Raj et al. [46] have introduced the concepts of fuzzy prime ideals of an ADL and studied them deeply.

In this paper, we introduce a topology on the set of all -fuzzy prime ideals of an ADL and the resulting space is called the -fuzzy prime spectrums of , denoted by or . For an -fuzzy ideal of , open subset of is of the form and is a closed set. In particular, we prove that is a base for a topology on . Furthermore, it is proved that the space is compact and it contains a subspace homeomorphic with the spectrum of which is dense in it. Also, it is proved that the space is a Hausdorff space if and only if the space is a -space and, further, it is noted that the space is a -space if and only if every -fuzzy prime ideal is an -fuzzy maximal ideal and -fuzzy minimal prime ideal of . Finally, it is proven that if and are isomorphic ADLs, then the space is homeomorphic with the space .

Throughout this paper, stands for an ADL with a maximal element and stands for a complete lattice satisfying the infinite meet distributive law; that is, and .

#### 2. Preliminaries

In this section, we recall some definitions and basic results mostly taken from [2, 4].

Definition 1. An algebra of type is called an Almost Distributive Lattice (abbreviated as ADL) if it satisfies the following conditions for all and :(1)(2)(3)(4)(5)(6)Any bounded below distributive lattice is an ADL. Any nonempty set can be made into an ADL which is not a lattice by fixing an arbitrarily chosen element in and by defining the binary operations and on by This ADL is called a discrete ADL.

Definition 2. Let be an ADL. For any and , define if . Then is a partial order on with respect to which 0 is the smallest element in .

Theorem 1. The following hold for any and in an ADL :(1) and (2)(3)(4)(5)(6) (i.e., is associative)(7)(8)(9)(10)(11)(12)

Definition 3. Let be a nonempty subset of an ADL . Then is called an ideal of if and for all .
As a consequence, for any ideal of for all and . An element is said to be maximal if, for any , implies . It can be easily observed that is maximal if and only if for all .

Definition 4. Let and be lattices and let be a mapping. Then is called (1) an order homomorphism (or isotone) if in in and (2) a lattice homomorphism if, for any , and .

Theorem 2. Let and be lattices and let be a bijection. Then is a lattice isomorphism if and only if both and are order homomorphisms.

Definition 5. Let and be ADLs. A mapping is called a homomorphism if the following are satisfied for any and : (1) . (2) . (3) .

Definition 6. Let and be topological spaces and let be a mapping; then is said to be continuous if and only if inverse image of every open set in is open in .

Definition 7. Let and be topological spaces and let be a mapping; then is said to be open if and only if image of every open set in is open in .

Definition 8. Let and be topological spaces; then a bijection is said to be a homeomorphism if it is a continuous open mapping.

Definition 9. An -fuzzy subset of is said to be an -fuzzy ideal of , if and , for all .

Theorem 3. Let be an -fuzzy subset of . Then is an -fuzzy ideal if and only if , , and , for all .

Definition 10. Let denote the characteristic function of any subset of an ADL ; that is,

Definition 11. A proper -fuzzy ideal of is called an -fuzzy prime ideal of if, for any , or .

Theorem 4. Let be a proper -fuzzy ideal of . Then the following are equivalent to each other: (1) For each , or is a prime ideal of . is an -fuzzy prime ideal of . (3) For any , and either or .

Theorem 5. Let be an -fuzzy prime ideal of and let 0 be a prime element in . Then is an -fuzzy minimal prime ideal of if and only if is a minimal prime ideal of , for all .

Definition 12. A proper -fuzzy ideal of is called an -fuzzy maximal ideal of if, for each , either or is a maximal ideal of .

#### 3. Topological Space on -Fuzzy Prime Ideals

In this section, we introduce the Zariski topology on the set of -fuzzy prime ideals of an Almost Distributive Lattice . Our definition of -fuzzy prime ideal offers us an appropriate setting to introduce a topology on the set of -fuzzy prime ideals of . A topology is introduced on the set of all -fuzzy prime ideals of to obtain the space called the hull-kernel topology on the set of all -fuzzy prime ideals and denoted by or . First, we have the following.

Theorem 6. Let and be ADLs. Let be a lattice homomorphism and . If and are -fuzzy ideals of and , respectively, then is an -fuzzy ideal of , is an -fuzzy ideal of if is an epimorphism, and .

Proof. Define and as for each and for each . Then,(1), as is an -fuzzy ideal of . Let . Then,Also, let in . As is homomorphism, we get . But then (since is an -fuzzy prime ideal). That is, . Therefore, is antitone. Thus, is an -fuzzy ideal of .(2)Clearly, . Let be a lattice epimorphism. Let . Then there exists such that and . Thus, . Now,Thus, . Similarly, . Let in . Then, . Therefore, whenever and , we have . Now,This shows that . Thus, is an antitone map. Therefore, is an -fuzzy ideal of .(3)Let . Then,Thus, .

Theorem 7. If an -fuzzy subset of is an -fuzzy prime ideal of , then is a homomorphism from to .

Proof. As is an -fuzzy ideal, thenSince is an -fuzzy prime ideal of , we have or . In either case, we getAlso, and and is antitone (being an -fuzzy ideal) implying that and . Thus,From (8) and (9), we haveTherefore, from (7) and (10), is a homomorphism from to .
In Theorem 6, we have proved that inverse image of an -fuzzy ideal of an ADL is an -fuzzy ideal again. In the case of -fuzzy prime ideals, we have the following.

Theorem 8. Let and be ADLs. Let be a lattice homomorphism. If is an -fuzzy prime ideal of , then is an -fuzzy prime ideal of .

Proof. By Theorem 6, is an -fuzzy ideal of . Let . Then,Therefore, is an -fuzzy prime ideal of .

Theorem 9. Let and be ADLs. Let be a lattice isomorphism. If is an -fuzzy prime ideal of , then is an -fuzzy prime ideal of and .

Proof. By Theorem 6 (2), is an -fuzzy ideal of . Let . Then, and , for some . Therefore, . Now, if , then implies that (since is injective).
Therefore,Similarly, and if , then . Thus,Thus, is an -fuzzy prime ideal of . Also, let . Then,Therefore, .

Example 1. Consider the lattice whose Hasse diagram is given below:

and let and let and be binary operations on defined by
Then, is an ADL, which is not a lattice . Let be the closed unit interval of real numbers. Then is a frame with respect to the usual ordering. Define -fuzzy subsets and of and , respectively, by and , for all ; , , and ; and define a function by . Then, we observe that if and if . Thus, by Theorem 4, is an -fuzzy prime ideal of . Also, and are prime ideals of . Therefore, is an -fuzzy prime ideal of (by4). For any and ,Thus, is a lattice homomorphism. Also, . Thus, is isotone and, for each , there exists such that . Hence, is a bijection map. Therefore, is a lattice isomorphism. By the above Theorems 6 and 8, and are -fuzzy prime ideals of and , respectively, since and are -fuzzy prime ideals.
In the following, we obtain a topological space by introducing Zariski topology on the set of -fuzzy prime ideals of ADLs.

Definition 13. Let be a nontrivial ADL and let be the set of all -fuzzy prime ideals of . For any -fuzzy subset of , we defineThe complement of in = .
Now, we prove some properties of and .

Theorem 11. Let and be -fuzzy subsets of . Then, we have the following: (1) if , then and ; and ; and , where is the smallest -fuzzy ideal containing and .

Proof. (1)Therefore, . Also,Therefore, .(2)Similarly,Therefore, (3)Clearly , since and by (1). On the other hand, let . Then ; it follows that . Hence, . Therefore, . Also, clearly , since and by (1). On the other hand,It follows that . Therefore, .(4)Let . Then . Therefore, and (as is an -fuzzy prime ideal).

Remark 1. In general, equality does not hold in Theorem 7. Equality holds if and are crisp ideals of . The following example shows that, in a case of -fuzzy subsets of , equality does not hold even if and are -fuzzy ideals.

Example 2. Let and with and let and be binary operations on defined by
Then, is an ADL. Now define by and . Then , and are prime ideals of . Therefore, is an -fuzzy prime ideal of . Define and as , and . Clearly, and are -fuzzy ideals of and . Then, and . From this, . Also, and hence . Now, but . Hence, .
Recall that if is an ideal of , then the characteristic function of is an -fuzzy ideal of . For such -fuzzy ideals, we have the following.

Theorem 12. Let and be ideals of . Then, and .

Proof. (1)Since , . Then, by Theorem 7 (2), we have . On the other hand, let . Then, , and it follows that , for each . If and , then there exists such that and . But, as , then . As is an -fuzzy prime ideal of , then . It follows that . This implies that either or , which gives a contradiction with the choice of and . So, or . Therefore, or and hence .(2)Clearly, . On the other hand,Therefore, .

Corollary 1. For any subsets of an ADL and letting , (1)(2)

Theorem 13. For any subsets of an ADL , we have the following:(1)(2)(3) for every (4)(5)

Proof. Clearly . Let be an -fuzzy ideal of such that . Then, , for all . Now, we shall prove that . For any , we haveNow,Therefore, and hence . Therefore, . This shows that is the smallest -fuzzy ideal of containing . Thus, .

Theorem 14. If is a family of -fuzzy subsets of , then

Proof. This shows that .

Theorem 15. Let . Then the pair is a topological space.

Proof. Consider -fuzzy subsets of defined by and , for all . Then, . Therefore, . Also, . Therefore, . Let and be -fuzzy subsets of . Then, . Now,Also, let be nonempty collection of -fuzzy ideals of . Then, we have (by the above theorem). Now,Therefore, is a topological space.

Definition 14. The topological space , as in Theorem 15, is called -fuzzy prime spectrum of an ADL and is denoted by or .
In the following section, we introduce base of .

Definition 15. Let be a nontrivial ADL and let be the set of all -fuzzy prime ideals of . For any , we define . That is,and hence

Theorem 16. Let be the set of -fuzzy prime ideals of an ADL . Then the following hold for any and :(1)(2)(3)(4) is maximal element in (5)(6)

Proof. Let be a prime ideal of . Then, is an -fuzzy prime ideal of . Put . Then, . Therefore, , for every prime ideal of . Hence, . Therefore, . Conversely, if , then there is no ideal not containing 0. It follows that there is -fuzzy prime ideal not containing and hence . Therefore, .
(4) Suppose that is maximal in . Then, . Then, no proper -fuzzy ideal contains and hence . Conversely,Let be a prime ideal of . Then, is an -fuzzy prime ideal of and let . Then and hence . Therefore, is maximal.
(5)Thus, (since ).
(6)Thus, (since ).

Theorem 17. For any subset of an ADL and , we have the following:(1)(2)

Proof. (1). Then, and and, hence, , for some . Since . Let . Then, . Now, we shall prove that . Clearly, and hence . Therefore, and hence . On the other hand, let such that , for some . Then, for some . This implies that and hence . Thus, .(2)

Theorem 18. is a base for a topology on .

Proof. By Theorem 16 (4), , for any maximal element of . Therefore, . Let . Then, for any , we have