-Fuzzy Prime Spectrums of ADLs

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. [4–6] 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