Abstract
In this paper, we introduce the concept of -fuzzy semiprime ideal in a general poset. Characterizations of -fuzzy semiprime ideals in posets as well as characterizations of an -fuzzy semiprime ideal to be -fuzzy prime ideal are obtained. Also, -fuzzy prime ideals in a poset are characterized.
1. Introduction
Fuzzy set theory was introduced by Zadeh in 1965 as an extension of the classical notion of set theory [1]. In 1971, Rosenfeld wrote his seminal paper on fuzzy subgroups in [2]. This paper has provided sufficient motivations for researchers to study the fuzzy subalgebras of different algebraic structures, like rings, modules, vector-spaces, lattices, and more recently in MS-algebras, universal algebras, pseudocomplemented semilattice, and so on (see [3–26]).
Zadeh defined a fuzzy subset of a nonempty set as a function from to unit interval of real numbers. Goguen in [27] generalized the fuzzy subsets of , to L-fuzzy subsets, as a function from to a lattice . Swamy and Swamy [5] initiated that complete lattices satisfying the infinite meet distributivity are the most appropriate candidates to have the truth values of general fuzzy statements.
In the literature, we have found several types of ideals and filters of a poset which are generalizations of ideals and filters of a lattice (see [28–33]). Halaś and Rachůnek in [34] introduced the notions of prime ideals in a poset, and Khart and Mokbel [35] introduced the concept of a semiprime ideal in general poset.
In [36, 37], the authors of this paper introduced several types of -fuzzy ideals and filters of a partially ordered set whose truth values are in a complete lattice satisfying the infinite meet distributive law. In addition, in [38], we have introduced and presented certain comprehensive results on the notion of -fuzzy prime ideals and maximal -fuzzy ideals of a poset by applying the general theory of algebraic fuzzy systems introduced in [39, 40].
Initiated by the above ideas and concepts, in this paper, we introduce and develop the concepts of -fuzzy semiprime ideal in a general partially ordered set. Characterizations of -fuzzy semiprime ideals in posets as well as necessary and sufficient conditions of an -fuzzy semiprime ideal to be -fuzzy prime ideal are observed. Also, by introducing the concept of a -atom element in a poset, we obtain characterizations of -fuzzy semiprime ideals and -fuzzy prime ideals in a poset satisfying the descending chain condition (DCC).
2. Preliminaries
For the necessary concepts, terminologies, and notations, we refer to [41, 42].
A pair is called a partially ordered set or simply a poset if is a nonempty set and “” is a partial order on . An element is called a lower bound of if for all . An upper bound is defined dually. The set of all lower bounds of is denoted by and the set of all upper bounds of , by .
By the sets and , we mean and , respectively. For any , the sets and are denoted by and , respectively. Furthermore, for subsets of , is denoted by and the set is denoted by . Similar notations are used for the set of all upper bounds of subsets of a poset .
For any subsets of a poset , we note that and , and if in , then and . In addition, and . An element in is called the greatest lower bound of or infimum of S, denoted by , if and . Dually, we have the concept of the least upper bound of or supremum of S which is denoted by .
For , we write (read as “ meet ”) in place of if it exists and (read as “ join ”) in place of if it exists. An element in is called the smallest (respectively, the largest) element of a poset if (respectively, ) for all . The smallest (respectively, the largest) element if it exists in is denoted by 0 (respectively, by 1). A poset is called bounded if it has and .
A poset is is said to satisfy the ascending chain condition (ACC), if every nonempty subset of Q has a maximal element. Dually, we have the concept of descending chain condition (DCC) [35].
Definition 1. (see [43]). A poset is called distributive if for all ,
Definition 2. (see [24]). A subset of a poset is called an ideal in if whenever .
Now, we consider the concept of a semiprime ideal introduced by Khart and Mokbel in a poset and by Rav in a lattice, as given in the following.
Definition 3. (see [28]). A proper ideal of a poset is called a semiprime ideal of if for all and imply
Dually, we have the concept semiprime filter of a poset .
Definition 4. (see [44]). A proper ideal of a lattice is called a semiprime ideal of if for all and together imply
Dually, we have the concept semiprime filter of a lattice .
For an ideal and an element in a poset , define a set by
Definition 5. (see [28]). An element in a poset is called an -atom with respect to an ideal of if and for any with implies .
Throughout this paper, stands for a complete lattice satisfying the infinite meet distributive law and stands for a poset with .
By an -fuzzy subset of a poset , we mean a mapping from into . We denote the set of - fuzzy subsets of by . For each and , the -level subset of , which is denoted by , is a subset of given by For fuzzy subsets and of , we write to mean for all in the ordering of . It can be easily verified that “” is a partial order on the set and is called the point-wise ordering. We write if and .
The following notions and results in this section are from the authors’ work in [29, 31].
Definition 6. is said to be an - fuzzy semi-ideal of if and for any , , for all .
Definition 7. is said to be an - fuzzy ideal of if and, for any, ,An -fuzzy ideal of is called a --fuzzy ideal if, for any , there exists such that .
Lemma 1. is an -fuzzy ideal of if and only if is an ideal of , for all .
Lemma 2. If is an - fuzzy ideal of , then is anti-tone.
Note that, for any in the constant -fuzzy subset of which maps all elements of onto , is denoted by .
Definition 8. An -fuzzy ideal of a poset is called proper, if , where is the largest element in .
Definition 9. A proper -fuzzy ideal of a poset is called an -fuzzy prime, if, for any ,
Definition 10. A proper -fuzzy ideal of a poset is said to be maximal if is a maximal element in the set of all proper -fuzzy ideals of .
3. -Fuzzy Semiprime Ideals of a Poset
In this section, we introduce and develop the notions of L-fuzzy semiprime ideal of a poset and give several characterizations of it. We shall begin with its definition.
Definition 11. An -fuzzy ideal of a poset is called an -fuzzy semiprime ideal if for all ,The following result characterizes any -fuzzy semiprime ideal of Q in terms of its level subsets.
Lemma 3. An -fuzzy ideal of is an -fuzzy semiprime ideal of if and only if is a semiprime ideal of for all .
Proof. Suppose that is an -fuzzy semiprime ideal and . Then, clearly, is an ideal of . Let such that and and . Then, and . This implies thatTherefore, Since is an -fuzzy semiprime ideal and , we haveThis implies that for all and hence, . Therefore, is a semiprime ideal of a poset .
Conversely, suppose that is a semiprime ideal of for all . Then, clearly, is an - fuzzy ideal of . Let and putThen,That is, and . This implies that and . Thus, since is a semiprime ideal of , we have . Therefore, for all , and hence is an -fuzzy semiprime ideal of .
Corollary 1. A subset of a poset is a semiprime ideal of if and only if its characteristic map is an -fuzzy semiprime ideal of .
Definition 12. An -fuzzy ideal of a lattice is called an -fuzzy semiprime ideal, if for all ,Dually, we have the concept of -fuzzy semiprime filter of a lattice .
Lemma 4. Let be an -fuzzy ideal of . Then, for any ,whenever exists in .
The following theorem shows that an L-fuzzy semiprime ideal of a poset is a natural generalization of an L-fuzzy semiprime ideal of a lattice.
Theorem 1. Let be a lattice. Then, an -fuzzy ideal of is an -fuzzy semiprime ideal in the poset if and only if it is an -fuzzy semiprime ideal in the lattice .
Proof. Let be an -fuzzy semiprime ideal in the poset and . Then, since , we haveAgain, since , and is antitone, we clearly haveTherefore, is an -fuzzy semiprime ideal in the lattice .
Conversely, suppose that is an -fuzzy semiprime ideal in the lattice . Let and . Then, and for all . Since , we have This implies that and henceSo, is an -fuzzy semiprime ideal in the poset .
The following result establishes a connection between -fuzzy prime ideals and -fuzzy semiprime ideals of a poset .
Lemma 5. Every -fuzzy prime ideal of a poset is an -fuzzy semiprime ideal.
Proof. Let be an -fuzzy prime ideal of . Let . Then since is an -fuzzy prime ideal of , we clearly haveLet . Then, and . Now if or , then we haveAgain if and , then we haveNow since and is an -fuzzy ideal, we haveHence, in either cases, we haveHence, is an -fuzzy semiprime ideal of .
Remark 1. The converse of the above lemma is not true. For example, consider the poset depicted in Figure 1. Define a fuzzy subset byThen, is an -fuzzy semiprime ideal but not an -fuzzy prime ideal of . This is because and .
Now, given an -fuzzy ideal of a poset and any element in , we define the following -fuzzy subset of .
Definition 13. Let be an -fuzzy ideal of and . Define an -fuzzy subset of byFrom Definition 9, observe that an -fuzzy ideal of is an -fuzzy prime ideal if, for any ,Now, we have the following lemmas.
Lemma 6. Let be an -fuzzy ideal of and . Then, is an -fuzzy semi-ideal containing .
Proof. NowTherefore, . Again, let and . NowTherefore, is an -fuzzy semi-ideal. Again, for all , we haveHence, .
Note that, for any , observe that .
Remark 2. For an -fuzzy ideal of a poset need not be an -fuzzy ideal of for all . For example, consider the poset depicted in Figure 2.
Define a fuzzy subset byThen, is an -fuzzy ideal of , and is a fuzzy subset of given byObserve that but . This implies that is not an -fuzzy ideal of .
Lemma 7. Let be an -fuzzy ideal of a poset and . Then,
Proof. Now, we have
Lemma 8. Let be an -fuzzy ideal of a poset and . Then, the following hold:(1)(2)(3) if and only if
Proof. (1)Put and . Now we claim that . Let . Then, To prove the other side of the inequality, let . Then, Hence, the claim is true.(2)The proof is similar to 1.(3)Suppose . Then, , for all . Thus, in particular, :Conversely suppose that . Now since, for any , , we have for all . Therefore, .
Now, we present a characterization of an -fuzzy semiprime ideal of a poset in terms of where is an -fuzzy ideal of and .
Theorem 2. An -fuzzy ideal of a poset is an -fuzzy semiprime ideal if and only if is an -fuzzy ideal for all , in fact, an -fuzzy semiprime ideal for all .
Proof. Let be an -fuzzy semiprime ideal of and . First, let us show that is an -fuzzy ideal of . Since , we have . Again, let and . Then,Again, since implies , we have for all This implies thatTherefore, is an -fuzzy ideal of for all . Now, we show that is an -fuzzy semiprime ideal of . Let and . Now,Now, implies that for all . Thus, we haveThus, . Therefore, is an -fuzzy semiprime ideal of .
Conversely, suppose that is an -fuzzy ideal of for all . Now we show that is an -fuzzy semiprime ideal of . Let and . Then,This implies thatThus,Therefore, is an -fuzzy semiprime ideal of .
The next result is a characterization of an -fuzzy ideal to be an -fuzzy prime ideal in a poset .
Theorem 3. Let be a proper -fuzzy ideal of a poset . Then, is an -fuzzy prime ideal of if and only if for all such that .
Proof. Suppose that is an -fuzzy prime ideal of and let such that . Then, by Lemma 5 and Theorem 2, it is clear that is an -fuzzy ideal of . Now we claim that . Now, for any , we have . However, as is an -fuzzy ideal of , . Thus, for all and hence .
Conversely, suppose that the given condition holds. Let . Now, we claim thatSuppose that . Then, . This implies that . Thus, by hypothesis, we have and hence .
Therefore, is an -fuzzy prime ideal of .
Now before we prove some other characterizations of -fuzzy primeness and -fuzzy semiprimeness in the case of a poset satisfying DCC, we introduce the concept of a -atom of an -fuzzy ideal of a poset.
Definition 14. Let be an -fuzzy ideal of a poset and . An element in is called a -atom with respect to , if it satisfies the following conditions:(1)(2) whenever
Example 1. Consider the poset depicted in Figure 3. Define a fuzzy subset byThen, it is easy to see that is an -fuzzy ideal of and is a -atom with respect to in .
Lemma 9. There always exists a -atom for every proper -fuzzy ideal in a poset satisfying DCC with respect to some in .
Proof. Let be a poset satisfying DCC and be a proper -fuzzy ideal of . Then, there exists such that . This implies that there exists such that . Put . Then, since , is a proper ideal of and is a nonempty subset of Q. Since is satisfying DCC, has a minimal element, say , such that . Now we claim that is a -atom with respect to . Since , we have . Let . Then, by the minimality of , and hence . Hence, the claim is true.
Remark 3. Lemma 9 gives a guarantee that if is an -fuzzy ideal of a poset satisfying DCC and for some and , then there exists a -atom in with respect to such that .
Lemma 10. Any two distinct -atoms of an -fuzzy ideal of a poset with respect to are incomparable.
Proof. Let be an -fuzzy ideal of and and be any two distinct -atoms with respect to . Then, by definition, we have and whenever and and whenever . Now we show that and are incomparable. Suppose not. Then or , i.e., or , which is a contradiction to the fact that and . Hence, and are incomparable.
Remark 4. From Lemma 10, we can deduce that if and are -atoms in a poset with respect to some in such that , then .
Lemma 11. Let be an -fuzzy semiprime ideal of a poset satisfying DCC. Then, is a --fuzzy ideal for every -atom in with respect to in .
Proof. Let be a -atom in with respect to in . Since is an -fuzzy semiprime ideal, by Theorem 2, is an -fuzzy ideal of . Now we show that is a - -fuzzy ideal. Suppose on the contrary that is not a --fuzzy ideal. Then, there exist such thatThis implies that there exists such thatThus, by Remark 3, there exists a -atom, say , with respect to such that . Since and , we have and hence . This implies that . Again, let . Then, . Therefore, is also a -atom with respect to . Also since , by Remark 4, we have . This implies that and hence for all and so . Since is an ideal, we havewhich is a contradiction to the fact that . Therefore, is a --fuzzy ideal.
Theorem 4. Let be an -fuzzy ideal of a poset satisfying DCC. Then, is an -fuzzy semiprime ideal of if and only if is an -fuzzy ideal, in fact, an -fuzzy prime ideal of for every -atom with respect to in .
Proof. Let be an -fuzzy semiprime ideal of a poset satisfying DCC and is a -atom in Q with respect to in . Then, by Lemma 11, is a --fuzzy ideal. Now, we have to show that is an -fuzzy prime ideal. Since , by Lemma 8, . Hence, is proper. Let and suppose thatPut . Since , there exists in such that . Then, by Remark 3, there exists a -atom, say , in with respect to such that . It is also clear that is also a -atom with respect to . Since , by Remark 4, we must have , and therefore , i.e., . Thus, we haveThis proves that is an -fuzzy prime ideal for every -atom .
Conversely, suppose that is an -fuzzy ideal for any -atom with respect to in . Let . Now, we claim thatSuppose not. Then, there exists such thatHence, by Remark 3, there exists a -atom in with respect to in such that . Then, by hypothesis, is an -fuzzy ideal. Again, since and , we havewhich is a contradiction to the fact that is a -atom with respect to . Therefore, is an -fuzzy semiprime ideal of .
The following result gives another characterization for -fuzzy semiprime ideals to be -fuzzy prime.
Theorem 5. Every maximal -fuzzy semiprime ideal of a poset is an -fuzzy prime ideal.
Proof. Let be a maximal -fuzzy semiprime ideal of a poset , that is, maximal among all proper -fuzzy semiprime ideals of a poset . Let . Then, by Theorem 2, is an -fuzzy semiprime ideal. Since , by maximality of , we have either or . If , then, by Lemma 8, . Thus,Again if , then we have . Thus, in either cases, we haveHence, is an -fuzzy prime ideal of .
As a consequence, we have the following corollary.
Corollary 2. Let be a maximal -fuzzy ideal of . Then, is an -fuzzy semiprime ideal if and only if is an -fuzzy prime ideal.
The following is a characterization of an -fuzzy ideal to be -fuzzy prime ideal in terms of a -atom in a poset satisfying DCC.
Theorem 6. Let be an -fuzzy ideal of a poset satisfying DCC. Then, is an -fuzzy prime ideal if and only if has exactly one -atom with respect to some in .
Proof. Let be an -fuzzy prime ideal of a poset satisfying DCC. Since is proper, by Lemma 9, there exists a -atom in with respect to some in . Now, we claim that has exactly one -atom with respect to in . Suppose not. Let be any distinct -atoms in with respect to in . Then, by Lemma 10, are incomparable and for all and for all . This implies that . Since or , we have or , which is a contradiction. Therefore, has exactly one -atom with respect to in .
Conversely suppose that has exactly one -atom, say , with respect to some in . Now, we show that is an -fuzzy prime ideal. Since , we have and hence is proper. Now, we show that for any ,Suppose not. Thus, there exist such thatThen, there exist -atoms with respect to such that and . Then, by hypothesis, we have and hence . Therefore , which is a contradiction to the fact that is a -atom with respect to . Therefore, is an -fuzzy prime ideal.
Lemma 12. Let be a proper -fuzzy ideal of a poset satisfying DCC and . Then, .
Proof. We show that as the converse inclusion always holds. Suppose that . This implies that there exists such that . Thus, there exists a -atom with respect to such that .
Then, we have , and hence,which is a contradiction to the fact that is a -atom with respect to . Hence, . Therefore, .
Lemma 13. The intersection of any nonempty family of -fuzzy prime ideals of is an -fuzzy semiprime ideal .
Proof. Let be a nonempty family of -fuzzy prime ideals of . Put . Then, clearly, is an -fuzzy ideal of . Let and Now,This implies thatTherefore, is an -fuzzy semiprime ideal .
As an immediate consequence of Theorem 4, Lemma 12, and Lemma 13 in the case of posets satisfying DCC, we obtain the following result.
Theorem 7. Let be a proper -fuzzy ideal of a poset satisfying DCC. Then, is an -fuzzy semiprime ideal of if and only if is expressed as an intersection of -fuzzy prime ideals of .
In the following, we characterize the distributive posets in terms of -fuzzy semiprime ideals in the following results.
Theorem 8. A poset is distributive if and only if of is an -fuzzy semiprime ideal of , for each .
Proof. Suppose that is distributive poset and . Now to show is an -fuzzy semiprime ideal of , by Corollary 1, it is enough to show that is a semiprime ideal of . Let such that and . Let . Then, and . This implies thatSince , we have and .
This implies that . Thus, and hence, .
Therefore, is an -fuzzy semiprime ideal of .
Conversely, suppose that is an -fuzzy semiprime ideal of for each . Then, by Corollary 1, it is clear that is semiprime ideal of for each . Let . It is enough to prove that as the converse inclusion is always true. Now let and . We claim that . Indeed, since , we haveThen, by semiprimeness of , we conclude that . Hence, for all . Therefore, , and hence,This proves that is a distributive poset.
Note that in a distributive poset every -fuzzy ideal need not be an -fuzzy semiprime ideal. Consider the distributive poset depicted in Figure 3.
Define a fuzzy subset byThen, is an -fuzzy ideal but not an -fuzzy semiprime ideal as and butAs an immediate consequence of Theorem 7 and Theorem 8, we have the following corollary.
Corollary 3. Let be a poset satisfying DCC. Then, is distributive if and only if, for every , is representable as an intersection of -fuzzy prime ideals of .
4. Conclusion
In this work, we introduce the notions of -fuzzy semiprime ideal in general posets. Characterizations of -fuzzy semiprime ideals in posets as well as characterizations of an -fuzzy semiprime ideal to be -fuzzy prime ideal are obtained by introducing the concept -atom elements in a poset. Also, -fuzzy prime ideals in a poset are characterized. Our future work will focus on the relations between the -fuzzy semiprime (resp., -fuzzy prime) ideals of a poset and the -fuzzy semiprime (resp., -fuzzy prime) of the lattice of all ideals of a poset. We will also extend and prove an analogue of Stone’s theorem for finite posets using -fuzzy semiprime ideals.
Data Availability
No data were used to support the results of this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research work was partially supported by the research grant from the College of Science, Bahir Dar University.