Abstract
We study the space of prime fuzzy ideals (and the space of maximal fuzzy ideals as a subspace) equipped with the hull-kernel topology in partially ordered sets. Mainly, we investigate the conditions for which the fuzzy prime spectrum of a poset is compact, Hausdorff, and normal, respectively.
1. Introduction
Partially ordered sets (posets) play a fundamental role in various branches of mathematics, providing a rich framework for studying order relations and their associated structures. In 1995, Halaś [1] introduced the general theory of ideals in a partially ordered set as an abstraction of ideals in lattices. Other scholars have also been proposed different definitions for ideals in a poset. For instance, closed or normal ideals of a poset were introduced by Birkhoff in [2]. Later in 1954, Frink [3] proposed the definition of ideals in a poset using lower and upper cones. Venkatanarasimhan [4] has also developed the theory of semiideals in a poset which are known by the name down sets. In 1979, Enré [5] has come up with a new definition of ideals in a poset called -ideals, generalizing almost all definitions proposed by other scholars. In 1995, Halaś and Rachunek [6] have studied primness in the class of ideals of a poset. Further investigations have been conducted by Erné [7] providing several characterizing theorems for prime and maximal ideals. The theory of prime ideals has been also extended to the class of -normal posets by Halaš et al. [8] and to the class of -distributive lattices by Joshi and Mundlik [9].
In 2016, Mundlik et al. [10] studied the hull-kernel topology on prime ideals in posets. The hull-kernel topology is a well-known topology that provides a powerful tool for analyzing the convergence, compactness, and connectedness properties in the class of ideals.
In recent years, the study of fuzzy sets and their applications in algebraic structures, particularly in lattices and ordered algebras, has gained significant attention, leading to the exploration of fuzzy ideals and their properties. Fuzzy sets allow for the representation of partial membership, capturing the idea that elements can belong to a set to varying degrees. This concept has proven to be a powerful tool in modeling uncertainty and vagueness, making it particularly well-suited for analyzing posets where precise membership may not always be applicable. In this context, the notion of fuzzy ideals and filters in posets has been studied by Assaye Alaba et al. [11, 12] as a natural extension of fuzzy ideals in lattices. Fuzzy ideals provide a flexible framework for studying the behavior of fuzzy subsets that possess certain desirable properties within a poset. They offer a more comprehensive understanding of the structure and relationships between elements, allowing for a deeper exploration of the underlying order relations. They further investigate the primness and maximality conditions in the class of fuzzy ideals in a general poset in [13]. In addition, they obtained sufficient conditions for the existence of fuzzy prime ideals in a poset as in the lattice of its fuzzy ideals.
The space of fuzzy prime ideals, called the hull-kernel topology for fuzzy prime ideals, has been studied by many authors in different classes of general algebraic structures such as rings (see [14]), modules (see [15]), hemirings (see [16]), semirings and -Semirings (see [17]), distributive lattices (see [18]), -algebras (see [19]), etc. This topology provides a powerful tool to explore and hence to utilize the important topological properties that the class of fuzzy prime ideals possesses inherently.
The motivation behind this study lies in the desire to identify the natural properties of the fuzzy prime spectrum of a poset and to understand the conditions under which the fuzzy prime spectrum of a poset exhibits the well-known topological properties such as compactness, Hausdorffness, and normality. Compactness, for example, is a fundamental property in topology that has implications in various areas of mathematics, including measure theory and mathematical analysis. By characterizing the conditions under which the fuzzy prime spectrum is compact, we can identify posets where the space of fuzzy prime ideals behaves in a particularly well-behaved manner. Hausdorffness, another crucial topological property, ensures the separation of points in a space, allowing for distinctness and uniqueness. Investigating the conditions for Hausdorffness in the hull-kernel topology for fuzzy prime ideals provides insights into the relationships between different elements within the poset and sheds light on the structure of fuzzy prime ideals. Furthermore, the study of normality in the fuzzy prime spectrum of a poset is motivated by its fundamental role in topology. Normal spaces possess certain desirable properties, enabling the existence of certain types of continuous functions and preserving important topological properties. By examining the conditions for which the fuzzy prime spectrum becomes a normal space, we contribute to the understanding of the broader topological structure of fuzzy ideals in partially ordered sets.
It is known that the unit interval has so many important properties: lattice theoretic property as well as topological properties. For instance, it forms a complete residuated lattice with three standard product and residuum operations (Lukasiewicz operations, Godel operations and Goguen (product) operations). By abstracting its lattice properties, Goguen [20] was first to define the concept of -fuzzy sets by replacing the unit interval by a general complete lattice in the definition of fuzzy subsets. Swamy and Swamy [21] mentioned that complete Brouwerian lattices are the most appropriate candidates to have the truth values of general fuzzy statements. Taking this into consideration, we prefer complete Brouwerian lattices to be the set of truth degrees for our fuzzy statements.
2. Preliminaries
A partially ordered set (or shortly a poset) is a system consisting of a nonempty set together with a partial ordering “” on , where by a partial ordering on , we mean a subset of which is transitive , antisymmetric , and reflexive . If is a partial ordering on and , then we write instead of . For , an element in is said to be an upper bound (respectively, a lower bound) of if (respectively, ) for all . We will denote by (respectively, ) the set of all upper bounds (respectively, lower bounds) of . We write (respectively, ), to denote (respectively, ). For any , we will use standard notations and to denote the sets and . In a dual manner, and will be denoted by and , respectively. For sets and in a poset , one can easily verify that and . Moreover, . It is also obvious that and for all . For , its infimum (or greatest lower bound) is a lower bound of such that for all lower bounds of . Similarly, the supremum (least upper bound) of is an upper bound of such that for all upper bounds of . We write (respectively, ) to say that is the infimum (respectively, the supremum) of . For if the infimum (respectively, the supremum) of the set exists, then it will be denoted by (respectively, ). A poset is said to have the least (respectively, the greatest) element if there is an element in denoted by (respectively, ) such that (respectively, ) for all . By a bounded poset, we mean a poset having both the least and the greatest elements.
An element in a bounded poset is said to be complemented if there is such that . If every element of is complemented, then we say that is complemented. By a Boolean poset, we mean a distributive complemented poset. More details about Boolean posets can be found in [22].
An element in a poset with is said to be the pseudocomplement of , if , and for , implies . A poset with least element 0 is said to be pseudocomplemented if for all , there is such that and for , implies . In this case, is unique and is called the pseudocomplement of in [23]. Given a poset , by a -ideal of , we mean an ideal of such that for all [6]. In a dual manner, a filter of is called an -filter provided that for all [1]. A proper ideal (respectively, a proper filter ) of is called prime if for all , implies or [6].
3. Fuzzy Prime Spectrum of a Poset
The present paper is the continuation of Assaye Alaba et al.’s work [11–13], and so we will follow the same notations, definitions, and results on fuzzy ideals, -fuzzy ideals, fuzzy filters, -fuzzy filters, prime fuzzy ideals, prime fuzzy filters, maximal fuzzy ideals, and maximal fuzzy filters in a poset.
A fuzzy subset of is a fuzzy ideal (respectively, a fuzzy filter) if and for any , , for all .
A fuzzy ideal of is a -fuzzy ideal (respectively, an -fuzzy filter) if an for any , there exists such that .
For our work, we collect some useful notations from [10–13]:(1) denotes the family of all nontrivial complete Brouwerian lattices.(2)For , we denote by the set of all prime elements of .(3) denotes the collection of all posets.(4) denotes the collection of all posets with the least element .(5) denotes the collection of all posets with the top element .(6) denotes the collection of all bounded posets.(7)For and , we write to say that is an -fuzzy subset of (or simply a fuzzy subset if there is no confusion on ); that is, is a mapping.(8)For , is the set of normalized fuzzy subsets of ; that is, .(9)For , is the set of all fuzzy ideals of .(10)For , is the set of all -fuzzy ideals of .(11)For , is the set of all fuzzy filters of .(12)For , is the set of all -fuzzy filters of .(13)For , is the set of all prime fuzzy ideals of .(14)For , denotes the set of all prime fuzzy filters of .(15)For , is the set of all maximal fuzzy ideals of .(16)For , denotes the set of all maximal fuzzy filters of .(17)(18)(19)(20).
Definition 1. For , its generalized complement denoted by is a fuzzy subset of defined as follows:for all .
Lemma 2. Let and with for some . Then, if and only if .
Proof. Suppose that . Then, it was proved in [13] that the level set is a prime ideal of . In this case, is of the formfor all . Now, we fist show that . Clearly, we have Let and . If or , then it follows that . Otherwise, if and , then being a prime ideal, we get . So, there exists such that . Since , we have , and hence, . Thus,Therefore, . To show , let If and then Again, since is a prime ideal, we have So, there is such that and soLet or Then, since , we have , and hence, , for all Hence, in either cases, there exists such that . Thus, is an -fuzzy filter, and hence, .
Conversely, assume that . Then, by Theorem 3.3 of [13], it is enough to check that the set is a prime ideal of . It is clear that is a proper ideal of . Let such that . This implies that for all . Since is an -fuzzy filter, there exists such thatSo that . Since is prime and hence irreducible in , we have either or , i.e., either or . So, is a prime ideal. Hence, it is proved.
Lemma 3. Let and with for some . Then, if and only if .
Lemma 4. For any and , (the fuzzy filter generated by ) where is a fuzzy subset of called a fuzzy point of given bywhich belongs to the class .
The next lemma proves existence of prime fuzzy ideals in posets from .
Lemma 5. Let . If has dual atoms, then the class is nonempty.
Proof. Let and a dual atom in . Putwhere is a fuzzy filter of generated by the fuzzy point which is characterized byfor all . Now, since and , by Zorn’s lemma, there exists a maximal -filter such that . Define a fuzzy subset of Z byfor all Then, it is clear that is an -fuzzy filter containing and , and hence, . Thus, is nonempty and hence form a poset together with the inclusion ordering of fuzzy sets. Moreover, it can be easily verified that it satisfies the hypothesis of Zorn’s lemma. So, by applying Zorn’s lemma, we can find a maximal member, let say in . Now, we show that is maximal in the set of all -fuzzy filters. Let be a proper element in - such that Then, Since is a dual atom, we have either or If , then for all Hence, for all which contradicts to the fact that is proper. Therefore, , and also, it is clear that and so . Thus, by maximality of in , we get This proves that is maximal in -.
Due to the assumption that every maximal -fuzzy filter is maximal among all fuzzy filters, we have that is a maximal fuzzy filter. Further, as , so is a prime fuzzy filter. Thus, would be a two-valued fuzzy set and can be characterized asfor all where . It further yields that is a -fuzzy ideal. Moreover, is a prime fuzzy ideal and is described asfor all . Therefore, the set of all prime fuzzy ideals of is nonempty.
Let , and be any fuzzy subset of . Define sets by:
Lemma 6. Let . Then, for any :
Proof. Let . This implies that . Since , it is clear that , and hence, . Therefore, and again as , we clearly have . To show the other inclusion, let . Then, . Now, we claim that . Suppose that . This implies that , which is a contradiction. Hence, the claim is true. Therefore, , and hence, . Therefore, .
Theorem 7. The collection forms a topology on .
Proof. Since and , contains both and . Let .
Then, since , is closed under finite intersection. Further, let .
Then . Now we claim that . Let . Then, ; that is, . Since , we get so that . Hence, . To prove the other inclusion, let . Then, . This implies that so that μ ∈ , and hence, . Therefore, . Hence, is closed under arbitrary unions. Therefore, is a topological space.
The topological space is called the spectrum of prime fuzzy ideals of the poset , and it is denoted by .
Remark 8. For any fuzzy ideal of a poset , the sets and , respectively, are open sets and closed sets in the topological space .
Lemma 9. Let and . Then, there exists such that
Proof. Suppose that . Then, and . This implies that and . Thus, we have and . Since , where is a prime element, . This implies that and . But as is a prime ideal, we have , and hence, there exists such that . Thus, and and . By primness of , we have , and hence, , that is, , so there exists such thatTo show the other inclusion, let for some . Then, , that is, . This implies that and , otherwise if or , we have , which is a contradiction. Again, since , we clearly have and , and hence, and . Thus, and , and so we have and henceThis proves the lemma.
Corollary 10. Let be a meet semilattice. Then, for any and, we have
Lemma 11. Let . The collectionforms a basis for the open sets of .
Proof. Let be any open set in and . Then, . Then, there exists such that . If we put , we have and so that . Thus, the collection forms a basis for the open sets of .
For any subset of , denote , the closure of in , the smallest closed set containing .
Lemma 12. Let . For any , we have
Proof. Clearly, is a closed set of and . We show that any closed set containing also contains . Since is closed, by Remark 8, for some fuzzy ideal of . As , we have for every . Therefore, , and thus, . Hence, is the smallest closed set containing . Therefore, .
Definition 13. A topological space is defined to be(1)A -space if for any , there exists an open set in such that either and or and (2)A -space if for any , there exist two open sets and in such that either and or and
Theorem 14. is a -space for all .
Proof. Let be any two distinct points in . Then, either or . Assume without loss of generality . This implies that and . Thus, we get an open set containing but not . Therefore, is a -space.
Lemma 15. Let and be any two points in . Then, if and only if
Proof. Let . Then, for each neighborhood of Thus, for all Suppose that Then, which is a contradiction. Therefore,
Conversely, suppose that and be any neighborhood of Then, Since , we get , which gives that that is, Thus
In the following lemma, we obtain a set of equivalent conditions for which becomes a -space. Recall that a topological space is said to be a -space if is a closed set for any .
Lemma 16. For any , the following conditions are equivalent:(1) is a -space(2) for all (3) is antichain
Proof. : Let . Then, since is , we have , by Lemma 12. : Suppose that such that . Then, , and hence, . Therefore, is antichain. : Let then by Lemma 15. Since is an antichain, we have . therefore for each . Thus, is a -space.
Corollary 17. For a pseudocomplemented meet-semilattice , the following conditions are equivalent:(1) is a -space(2) for all (3) is antichain
Definition 18. We say a topological space reducible if there are two closed sets and such that and . is irreducible if it is not reducible. In particular, a closed subset of is irreducible if for any closed sets and in :
In the following lemma, we describe what irreducible sets look like in .
Lemma 19. Let and be closed in . Then, being a prime fuzzy ideal is a necessary and sufficient condition for to be irreducible.
Proof. Suppose that is irreducible. Put . Since , it is clear that is proper. Let and be fuzzy points in such that . Then, for all . Thus, for all , either or . Hence, . Since is irreducible and and are closed, we have either or and hence or . Hence, or . Therefore, is a prime fuzzy ideal of .
Conversely suppose that is a prime fuzzy ideal of . Let , where and are closed sets in . Put and . Thus, and . Also, observe that and since is a prime fuzzy ideal of , we have either or . Therefore, and . Then, by Lemma 12, we clearly have or .
Lemma 20. Let be a complemented poset. For , if we define , then is an antichain.
Proof. Let such that Suppose that Then, there exists such that and This implies that and By complementedness of , there exists such that Since is a prime ideal, , we have Thus, we have and hence which is a contradiction. Therefore, , and hence, is antichain.
Recall that can be made a subspace of X by the relativized topology where
If we put then It is evident that the familyconstitutes a base for
Corollary 21. Let be a complemented poset. If is an antichain, then is antichain.
Definition 22. A topological space is Hausdorff (or -space) if any two distinct points can be separated by means of disjoint open sets. That is, for any , there are two disjoint open sets and in such that and .
Theorem 23. Let . If is complemented, then(1) is both open and closed, for all such that (2) is a Hausdorff space
Proof. (i)We now claim that where is the complement of in Let Then, and so that . Hence, and . Since is a prime ideal, , we have Consequently, and , and thus, . On the other hand, let Then, and . As we have . It follows that , and hence, Otherwise, if we have which is a contradiction. Therefore, , and thus, Hence, this is the claim.(ii)Let . Without loss of generality, we can assume that . Since , there exists such that and Thus, and , and hence, and . Otherwise, if and we have and which is a contradiction. Whence and . Let such that Then, and Hence, and and for some . Since we have So, is Hausdorff.In a topological space , by a clopen set, we mean a set which is closed and open. We say that is totally disconnected if any two distinct points in can be separated by means of a clopen set, That is, for any , we can find a clopen set such that and .
Corollary 24. Let . If is complemented, then is totally disconnected.
4. Space of Maximal Fuzzy Ideals
This section is devoted to the study of the space of maximal fuzzy ideals of a poset as a subspace of the fuzzy prime spectrum under the assumption that every maximal fuzzy ideal is prime. Moreover, we derive equivalent conditions for the space of maximal fuzzy ideals of a poset to be a normal space.
Remark 25. Note that if and has dual atoms, then by applying Zorn’s Lemma, we can find a maximal fuzzy ideal in . Therefore, and so to study as a subspace of , first we have to make sure that the inclusion holds. Note that if , then . Thus, it follows from the assumption that forms a subspace of . Any open set of is of the form . For every fuzzy ideal of if we considerthen we have the following:
Lemma 26. Let . Then, the family forms a basis for as a subspace of .
Lemma 27. Let , and . Then, (the fuzzy ideal of generated by the fuzzy point ) is a -fuzzy ideal of .
Lemma 28. Let , in and such that . If every maximal element in is maximal in , then for every , is maximal in if and only if for any such that there exists such that and .
Proof. Let be a maximal -fuzzy ideal of and .
Consider the fuzzy subset defined by, for any We show that is a fuzzy ideal. Since it is clear that and let and . Now,Thus, . Since for any , we clearly have , and hence, . Again since , we clearly have , and hence, . This shows that . It follows that for some ; that is, . Again, by the hypothesis “every maximal element of is maximal in ,” would be maximal, and hence, . Since we have , and hence, there exist such that and . Then, . But, as , there should be some such that , and hence, . Therefore,
Conversely, assume the condition to be true and let such that . Then, there exists such that and Then, by hypothesis, there exits such that and This implies that and since , we have , and hence, Hence, is a maximal -fuzzy ideal.
Let us recall a dually dense element in a poset. Let and . We say that is dually dense whenever , where We denote by , the set of all dually dense elements in .
Lemma 29. Let . Assume that every maximal element of is maximal in the class . Then,
Proof. Let be the collection of all maximal -fuzzy ideals of . Suppose Then, that is . Suppose on the contrary that . Then, there exists a maximal -fuzzy ideal such that . By Lemma 28, there exists such that and This gives q ∈ , and hence, which is a contradiction to the fact that . On the other hand, let Then, for all maximal -fuzzy ideals. Suppose Then, , and hence, there exists such that . By Zorn’s lemma, there is a maximal -fuzzy ideal with . As and we have , and this implies that is a contradiction to the maximality of .
Corollary 30. Let and assume that that every maximal element of is maximal in the class . Then, is a fuzzy ideal of .
Lemma 31. Let . Then, the closure of the set in is .
Proof. It follows from Lemma 12 thatHence, this is proved.
Definition 32 (see [24]). By -space, we mean a topological space for which each of its nonempty open set contains a nonempty closed set.
One can easily verify that every -space can be viewed as a -space. One of our first aims here is to provide an example of a poset for which its fuzzy prime spectrum is not a -space. For, consider the poset whose Hasse diagram is depicted in Figure 1 and where . Here, . The collection of open sets is
Now, and so that is a nonempty open set which does not contain a nonempty closed set. Thus, is not , and hence, it is not a -space, as every -space is a -space.
Lemma 33. Let with the least element 0. Ifthen is a -space.
Proof. Suppose that . By Lemma 29, Let be any nonempty proper open set in . Clearly, Then, there exists a nonzero such that . Let say . Since is nonzero, we have By Lemma 29, . If for all , then which is a contradiction. Thus, there is a maximal -fuzzy ideal such that , and thus, . Since , it follows that is a prime fuzzy ideal. Further, yields, and hence, is a closed set. Therefore, is the desired nonempty closed set such that , and consequently, is a space.
Remark 34. But, the converse of Lemma 33 need not necessarily be true in general. Consider the poset whose Hasse diagram is given in Figure 2 and let , where . Here, . By Lemma 16, is a -space, and hence, it is , but the intersection of all maximal fuzzy ideals of is nonzero.
Definition 35. Given two subsets and of a topological space , we say that is weakly separable from if .
Theorem 36. Let . Let us putThen, is the least member of .
Proof. We first show that . For, let be any element outside of ; that is, . By Lemma 12, . As , there exists a maximal ideal By the hypothesis, is a prime fuzzy ideal. Thus, , and hence, . This implies , and therefore, is not weakly separable from any element outside it. Next, we show that for all . Let and be a prime fuzzy ideal of such that . Then, . So, by Lemma 12 there exists such that . As , it should be the case that which implies that is not maximal; i.e., leads to a conclusion . Therefore, is the least member in .
Lemma 37. Let be ideal distributive and be such that for any , there exists such that For and , if for some index set , then .
Proof. Put . Assume on contrary that . Then, by our assumption, there exists a prime element such that . As is ideal distributive, need to necessarily have prime ideals and let be a prime ideal in . Define as follows:for all . Then, it is clear that μ ∈ , and hence, for some and . This implies that However, . Thus, is a contradiction. Therefore, .
Corollary 38. Let be such that for any , there exists a dual atom such that Then, for any and and any subspace of containing if for some index set , then .
Lemma 39. Let be a poset, , and Then,
Proof. The proof is similar to that of Proposition 16 in [25].
We now turn our attention and go to study some compactness properties of the space of prime (respectively, maximal) fuzzy ideals in a poset.
Lemma 40. Let be such that for any α ∈ , there exists such that Let and be any subspace of containing . Then, is compact.
Proof. Let be any subspace of containing and let be a basic open cover for and let Then, by Corollary 38 and Lemma 39, we clearly haveThen, we have . This implies that We claim that Suppose that Then, by Zorn’s lemma, there exists a maximal fuzzy ideal in such that Thus, a contradiction to the fact that . Hence, the claim holds true. This implies that the set . It was proved in [1] that the class of all ideals of forms an algebraic lattice with respect to the inclusion order and every principal ideal is a compact element in . Then, it follows from this fact that there exist such that Now, we show that . Let . Then, So, for each , and hence, i.e., for all This implies thatwhich is a contradiction. Thus, . So,
Corollary 41. For any , both and are compact.
Corollary 42. If is a bounded pseudocomplemented semilattice, then is compact. In particular, is compact.
Theorem 43. Let . Then, is Hausdorff if and only if for any distinct elements , we can find and such that and
Proof. Suppose that is Hausdorff. Let be any two distinct elements of . As is Hausdorff, we get two fuzzy ideals and of such that and are disjoint open neighborhoods of and , respectively. This gives that and so that there are two points with and Put and . Also, fromwe get . Therefore, This gives for all μ ∈ , and hence, . Now, let if , then it holds thatIf and , thenOn the other hand, if we are assuming that , then it is clear that .
Conversely, let and be any two distinct maximal fuzzy ideals in . By our assumption, there exist and such that and . Clearly, and are the neighborhoods of and , respectively, in . As , we haveThis shows that is Hausdorff. □
Definition 44. A bounded poset is said to have the -property if for any , there are and such that and .
Lemma 45. Let . If has the -property, then for each θ ∈ , there is a unique such that .
Proof. Let . Since is given to be bounded and hence with top element , we can find containing . To prove the uniqueness part, we use contradiction. Suppose on the contrary that is contained in two distinct maximal fuzzy ideals and . Since has the -property, there exist and such that and . As , we have In either case, we get a contradiction. Therefore, every prime fuzzy ideal is contained in a unique maximal fuzzy ideal.
We say that topological space is normal provided that any two disjoint closed sets in can be separated by means of disjoint open sets. That is, for any disjoint closed sets and in , there exist two disjoint open sets and in such that and . Observe that if is compact and Hausdorff, then it is normal.
Lemma 46. Let . If has the -property, then is Hausdorff. Moreover, it is normal.
Proof. Let and be two distinct distinct elements of . Since is a -poset, there exist and such that and . By Theorem 43, is Hausdorff. By Corollary 41, is a compact space. Thus, is a normal space.
The converse of the above lemma is true if the intersection of all maximal fuzzy ideals of is .
Theorem 47. Let with the bottom element 0. Assume that . Then, is Hausdorff if and only if has the -property.
Proof. It follows from Theorem 43. If is Hausdorff, there exist and such that and , for any distinct maximal ideals and . Thus, is a -poset. The converse follows from Lemma 46.
We conclude this paper by the following theorem:
Corollary 48. Let with the least element 0 and assume that . Then, the following conditions are equivalent:(1) has the -property(2)For any , we can find two open neighborhoods and of and in such that (3) is normal In addition, if is given to be complemented and is antichain, then the following statement is equivalent with any one (and hence all) of the above three statements(4) is a normal space
It is observed that if is a bounded distributive lattice, then the intersection of all prime fuzzy ideals of becomes . Thus, we can deduce the following corollary from the proof of Corollary 48.
Corollary 49. The following conditions are equivalent for any bounded distributive lattice :(1) has the -property(2)For any , we can find two open neighborhoods and of and in such that (3) is normal In addition, if is a Boolean algebra, then the following statement is equivalent with any one (and hence all) of the above three statements(4) is a normal space
5. Conclusions
This manuscript presents results on fuzzy prime ideals of partially ordered sets, focusing on the study of their topological properties within the context of the hull-kernel topology. Our investigation has centered around the space of prime fuzzy ideals, along with the space of maximal fuzzy ideals as a subspace. We explored the conditions for which the space of fuzzy prime ideals in partially ordered sets is compact, Hausdorff, and normal through which it deepened our understanding of the interplay between order relations and fuzzy ideals. We believe that these findings not only contribute to the field of fuzzy set theory but also provide valuable insights into the broader study of partially ordered sets and their associated structures.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All the authors contributed equally to the writing of this manuscript. They have also read and approved the final manuscript.
Acknowledgments
The authors of this paper would like to thank the Vice President’s office for research and technology transfer, Dilla University. This project was financially supported by Dilla University.