-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 haveThus, (since ). Therefore, form a base for a topology on .

Theorem 19. Let be an ADL and . Then, any closed subset of is of the form for some ideal of and any open subset of is of the form for some ideal of .

Proof. For each , is open in and and hence is open in . Since , it follows that is closed in . On the other hand, let be a closed subset of . Then, is open in and hence for some subset of , since is a base for a topology on . If , the ideal generated by in , then . Now, we shall prove that . Let . Then,On the other hand,Therefore, and hence .

Theorem 20. Let be an ADL and . Then, for any , is a compact open subset of if and only if for some .

Proof. Suppose that is compact open. Since is open, for some . Also, is compact and is a cover of . Then, there exists such that , where .
Conversely, suppose that for some . Therefore, is open. Now, we prove that is compact. Suppose that Then ; otherwise, if , then there exists a prime ideal of such that and . Let . Then, is an -fuzzy prime ideal of , since is prime and also . Since , it follows that and hence but , for every , which is a contradiction to our assumption. Therefore, and hence . This implies that . Thus, is compact.

Corollary 2. Every basic open set in is compact.

Corollary 3. Let be an ADL. Then, is compact if and only if has a maximal element.

Proof. Let . Suppose that is compact. Then, for some (by Corollary 2) and hence is maximal element in . On the other hand, if has a maximal element, say , then , which is compact (by 2).

Theorem 21. For any subset of , is closed in if and only if there exists such that .

Proof. Suppose that is closed in . Then, is open in andTherefore, , for some . Conversely, suppose that , for some subset of . Then, and hence is closed in , as is closed in . Therefore, the subset of is closed in .

Theorem 22. Let denote the set of all prime ideals of . Then the set is dense in .

Proof. Let be a prime ideal of .Then is an -fuzzy prime ideal of . Then . Let . Let be a basic open subset of containing . Now,Thus, , since is a prime ideal of . Therefore, contains a point of . Thus, every member of is a limit point of . Thus, .

Theorem 23. For any subset of , , where .

Proof. Let . Then,Therefore, . Thus, . But then is closed set containing . Hence, . On the other hand, let . Let . Let be a basic open set containing . Then,Thus, . This shows that is a limit point of . Thus, . Therefore, .

Corollary 4. For any subset , for any if and only if .

First, let us recall that a topological space is said to be space if, for any there exists an open set containing and not containing or vice versa, that is called a space if, for any , there exists open sets and in such that and , and that is called a Hausdorff space (space) if, for any , there exist disjoint open sets and in containing and , respectively.

Theorem 24. For any ADL , is a space.

Proof. Let . That is, and are distinct -fuzzy prime ideals of . Then, either or . Suppose that . Then, there exists such that . Then and . Also, each is open set containing but not . On the other hand, if , then there exists an open set containing but not . Therefore, is a space.

Theorem 25. Let be an ADL with maximal elements. Then the following are equivalent to each other:(1) is a space (or a Hausdorff space)(2) is a space(3)Every -fuzzy prime ideal of is an -fuzzy maximal ideal of (4)Every -fuzzy prime ideal of is an -fuzzy minimal prime ideal of

Proof. (1): clear.(2): suppose that is a space and is an -fuzzy prime ideal of . Since is closed in , . By Theorem 19, we have , which implies that there is no -fuzzy prime ideal of containing other than itself. That is, is an -fuzzy maximal ideal of .(3): suppose that every -fuzzy prime ideal of is an -fuzzy maximal ideal of . Suppose that is an -fuzzy prime ideal of . Let be an -fuzzy prime ideal of such that . By assumption, . Therefore, is an -fuzzy minimal prime ideal of .(4): suppose that every -fuzzy prime ideal of is an -fuzzy minimal prime ideal of . Let . That is, and are distinct -fuzzy prime ideals of . Then, either or . Suppose that . Then, there exists such that . Then, and . Also, each is open set containing but not . On the other hand, if , then there exists an open set containing but not . Hence, and . By the minimality of , there exists such that . Therefore, . Hence, is a Hausdorff space.

Theorem 26. Let and be ADLs and let be a lattice homomorphism. Define by , for each . Then, (1) is continuous mapping. (2) If is onto, then is one-to-one.

Proof. Clearly is a well-defined map.(1)Let be any basic closed set in and . Then, Therefore, inverse image under of any basic open set in is a closed set in and hence is continuous.(2)Let be onto and such that . Then, for each , Therefore, is a one-to-one map.

Theorem 27. Let be an isomorphism. Then, the space is homeomorphic with the space .

Proof. Let be an isomorphism. Define the function by , for each , and by , for each . Then, and are well defined and inverse of each other (by Theorems 8 and 9). Thus, by the above theorem, and are continuous. Therefore, and are homeomorphisms (by Definition 12).

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Conflicts of Interest

The authors declare that they have no conflicts of interest.